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Derivatives
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Logarithmic Differentiation
3 Activities
Wize University Calculus 1 Textbook > Derivatives
Derivatives of Inverse Trig Functions
3 Activities
Wize University Calculus 1 Textbook > Derivatives
The Chain Rule
3 Activities
Wize University Calculus 1 Textbook > Derivatives
The Quotient Rule
3 Activities
Find the derivative of the following functions at the given values
a)
f
(
x
)
=
arcsin
(
x
2
)
x
3
−
1
f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}
f
(
x
)
=
x
3
−
1
a
r
c
s
i
n
(
x
2
)
at
x
=
0
x=0
x
=
0
b)
g
(
x
)
=
(
cos
x
)
e
x
g\left(x\right)=\left(\cos x\right)^{e^x}
g
(
x
)
=
(
cos
x
)
e
x
at
x
=
0
x=0
x
=
0
Part a)
Part b)
I don't know
Check Submission
More Logarithmic Differentiation Questions:
Derivatives
Find the derivative of the following functions at the given values
a)
f
(
x
)
=
arcsin
(
x
2
)
x
3
−
1
f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}
f
(
x
)
=
x
3
−
1
a
r
c
s
i
n
(
x
2
)
at
x
=
0
x=0
x
=
0
b)
g
(
x
)
=
(
cos
x
)
e
x
g\left(x\right)=\left(\cos x\right)^{e^x}
g
(
x
)
=
(
cos
x
)
e
x
at
x
=
0
x=0
x
=
0
Logarithmic Differentiation
If
f
(
x
)
=
x
x
f(x)=x^{\sqrt{x}}
f
(
x
)
=
x
x
then find
f
′
(
4
)
f'(4)
f
′
(
4
)
.
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(
sin
(
x
y
)
)
x
=
x
1
/
4
(\sin(xy))^x = x^{1/4}
(
sin
(
x
y
)
)
x
=
x
1/4
at the point
(
1
2
,
π
2
)
\left(\dfrac{1}{2},\dfrac{\pi}{2}\right)
(
2
1
,
2
π
)
.
Logarithmic Differentiation
Use logarithmic differentiation to find
g
′
g'
g
′
where
g
(
x
)
=
x
x
2
2
+
tan
x
g(x)=x^{x^2}\sqrt{2+\tan x}
g
(
x
)
=
x
x
2
2
+
tan
x
Logarithmic Differentiation
If
f
(
x
)
=
(
2
x
)
sin
x
f(x)=(2x)^{\sin x}
f
(
x
)
=
(
2
x
)
s
i
n
x
, find
f
′
(
π
2
)
f'(\frac{\pi}{2})
f
′
(
2
π
)
?
Logarithmic Differentiation
If
f
(
x
)
=
x
x
f(x)=x^{\sqrt{x}}
f
(
x
)
=
x
x
then find
f
′
(
4
)
f'(4)
f
′
(
4
)
.
Logarithmic Differentiation
Use logarithmic differentiation to find
g
′
g'
g
′
where
g
(
x
)
=
x
x
2
2
+
tan
x
g(x)=x^{x^2}\sqrt{2+\tan x}
g
(
x
)
=
x
x
2
2
+
tan
x
Complicated Log Derivative
If
f
(
x
)
=
(
sin
x
)
3
x
f\left(x\right)=\left(\sin x\right)^{3x}
f
(
x
)
=
(
sin
x
)
3
x
, then
f
′
(
x
)
=
f'\left(x\right)=
f
′
(
x
)
=
Logarithmic Differentiation: Big Product/Quotient
Consider
y
=
(
x
2
+
1
)
2
(
x
−
1
)
4
(
x
3
+
2
)
3
3
\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}
y
=
3
(
x
3
+
2
)
3
(
x
2
+
1
)
2
(
x
−
1
)
4
. Compute
d
y
d
x
at
x
=
0
\displaystyle\frac{dy}{dx}\text{ at }x=0
d
x
d
y
at
x
=
0
.
Logarithmic Differentiation
Find the derivative of
y
=
(
x
+
1
)
(
x
−
1
)
y=\left(x+1\right)^{\left(x-1\right)}
y
=
(
x
+
1
)
(
x
−
1
)
Logarithmic differentiation
Find the derivative of the function
y
=
sin
(
x
)
sin
(
x
)
y = \sin(x)^{\sin(x)}
y
=
sin
(
x
)
s
i
n
(
x
)
Logarithmic Differentiation
Find the derivative of
y
=
(
x
+
1
)
(
x
−
1
)
y=\left(x+1\right)^{\left(x-1\right)}
y
=
(
x
+
1
)
(
x
−
1
)
Complicated Log Derivative
If
f
(
x
)
=
(
sin
x
)
3
x
f\left(x\right)=\left(\sin x\right)^{3x}
f
(
x
)
=
(
sin
x
)
3
x
, then
f
′
(
x
)
=
f'\left(x\right)=
f
′
(
x
)
=
Logarithmic Differentiation
Find the derivative of
y
=
(
x
+
1
)
(
x
−
1
)
y=\left(x+1\right)^{\left(x-1\right)}
y
=
(
x
+
1
)
(
x
−
1
)
Logarithmic Differentiation
Find the derivative of
y
=
(
x
+
1
)
(
x
−
1
)
y=\left(x+1\right)^{\left(x-1\right)}
y
=
(
x
+
1
)
(
x
−
1
)
Logarithmic differentiation
Find the derivative of the function
y
=
sin
(
x
)
sin
(
x
)
y = \sin(x)^{\sin(x)}
y
=
sin
(
x
)
s
i
n
(
x
)
Logarithmic Differentiation
Find the derivative of
y
=
(
x
+
1
)
(
x
−
1
)
y=\left(x+1\right)^{\left(x-1\right)}
y
=
(
x
+
1
)
(
x
−
1
)
Complicated Log Derivative
If
f
(
x
)
=
(
sin
x
)
3
x
f\left(x\right)=\left(\sin x\right)^{3x}
f
(
x
)
=
(
sin
x
)
3
x
, then
f
′
(
x
)
=
f'\left(x\right)=
f
′
(
x
)
=
Complicated Log Derivative
If
f
(
x
)
=
(
sin
x
)
3
x
f\left(x\right)=\left(\sin x\right)^{3x}
f
(
x
)
=
(
sin
x
)
3
x
, then
f
′
(
x
)
=
f'\left(x\right)=
f
′
(
x
)
=
Complicated Log Derivative
If
f
(
x
)
=
(
sin
x
)
3
x
f\left(x\right)=\left(\sin x\right)^{3x}
f
(
x
)
=
(
sin
x
)
3
x
, then
f
′
(
x
)
=
f'\left(x\right)=
f
′
(
x
)
=
Logarithmic Differentiation
Find the derivative of
y
=
(
x
+
1
)
(
x
−
1
)
y=\left(x+1\right)^{\left(x-1\right)}
y
=
(
x
+
1
)
(
x
−
1
)
Logarithmic differentiation
Find the derivative of the function
y
=
sin
(
x
)
sin
(
x
)
y = \sin(x)^{\sin(x)}
y
=
sin
(
x
)
s
i
n
(
x
)
Practice: Log Differentiation
Practice: Log Differentiation
Find the derivative of
x
2
(
cos
x
)
(
e
2
x
)
2
x
ln
(
x
)
\frac{x^2\left(\cos x\right)\left(e^{2x}\right)}{2x\ln\left(x\right)}
2
x
l
n
(
x
)
x
2
(
c
o
s
x
)
(
e
2
x
)
Complicated Log Derivative
If
f
(
x
)
=
(
sin
x
)
3
x
f\left(x\right)=\left(\sin x\right)^{3x}
f
(
x
)
=
(
sin
x
)
3
x
, then
f
′
(
x
)
=
f'\left(x\right)=
f
′
(
x
)
=
Logarithmic Differentiation
Use logarithmic differentiation to find
g
′
g'
g
′
where
g
(
x
)
=
x
x
2
2
+
tan
x
g(x)=x^{x^2}\sqrt{2+\tan x}
g
(
x
)
=
x
x
2
2
+
tan
x
Logarithmic Differentiation
Find
y
′
y'
y
′
for the function
𝑦
=
𝑥
2
−
x
2
𝑦 = 𝑥^{2-x^2}
y
=
x
2
−
x
2
.
Find the derivative of
y
=
(
cos
x
)
5
x
2
y=\left(\cos x\right)^{\frac{5x}{2}}
y
=
(
cos
x
)
2
5
x
Logarithmic for Implicitly Defined Function
Find
d
y
/
d
x
dy/dx
d
y
/
d
x
at the point
(
1
,
1
)
(1,1)
(
1
,
1
)
given
x
y
=
y
x
x^y=y^x
x
y
=
y
x
Logarithmic Differentiation: Big Product/Quotient
Consider
y
=
(
x
2
+
1
)
2
(
x
−
1
)
4
(
x
3
+
2
)
3
3
\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}
y
=
3
(
x
3
+
2
)
3
(
x
2
+
1
)
2
(
x
−
1
)
4
. Compute
d
y
d
x
at
x
=
0
\displaystyle\frac{dy}{dx}\text{ at }x=0
d
x
d
y
at
x
=
0
.
Logarithmic with Chain
Use logarithmic differentiation to find
g
′
g^\prime
g
′
where
g
(
x
)
=
x
x
2
+
sin
x
\displaystyle g(x)=x^x\sqrt{2+\sin{x}}
g
(
x
)
=
x
x
2
+
sin
x
.
Differentiation: Logarithmic with Trig
If
y
=
sin
x
x
y=\sin^xx
y
=
sin
x
x
, find
y
′
y'
y
′
.
Derivatives: Logarithmic Functions
Compute the derivative of
f
(
x
)
=
x
x
+
1
f(x) = x^{x + 1}
f
(
x
)
=
x
x
+
1
. Remember that
log
x
=
log
e
x
=
ln
x
\log x = \log_e x = \ln x
lo
g
x
=
lo
g
e
x
=
ln
x
.
Logarithmic Differentiation
Find the derivative of
y
=
(
x
+
1
)
(
x
−
1
)
y=\left(x+1\right)^{\left(x-1\right)}
y
=
(
x
+
1
)
(
x
−
1
)
Implicit Differentiation: Logarithmic Functions
For the following equations, solve for
d
y
d
x
\frac{dy}{dx}
d
x
d
y
:
a)
y
=
(
5
x
2
+
3
)
sin
(
x
)
y = (5x^2 + 3)^{\sin(x)}
y
=
(
5
x
2
+
3
)
s
i
n
(
x
)
b)
x
2
y
+
3
y
2
x
2
=
x
+
y
x^2y + 3y^2 x^2 = x + y
x
2
y
+
3
y
2
x
2
=
x
+
y
Logarithmic Differentiation
Calculate the derivative of the following functions.
f
(
x
)
=
(
tan
x
)
sec
x
\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
f
(
x
)
=
(
tan
x
)
sec
x
g
(
x
)
=
(
x
tan
4
x
)
x
g(x)=(x\tan{4x)}^x
g
(
x
)
=
(
x
tan
4
x
)
x
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(
sin
(
x
y
)
)
x
=
x
1
/
4
(\sin(xy))^x = x^{1/4}
(
sin
(
x
y
)
)
x
=
x
1/4
at the point
(
1
2
,
π
2
)
\left(\dfrac{1}{2},\dfrac{\pi}{2}\right)
(
2
1
,
2
π
)
.
Logarithmic Differentiation
Find the derivative of
f
(
x
)
=
(
tan
x
)
ln
x
f(x)=(\tan x)^{\ln x}
f
(
x
)
=
(
tan
x
)
l
n
x
Logarithmic Differentiation
Use logarithmic differentiation to find
g'
where
g
(
x
)
=
x
x
2
+
sin
x
g(x)=x^x\sqrt{2+\sin x}
g
(
x
)
=
x
x
2
+
sin
x
Logarithmic Differentiation
Compute the derivative of
f
(
x
)
=
x
3
/
2
x
+
1
(
x
+
5
)
7
sin
x
f(x)=\frac{x^{3/2}\sqrt{x+1}}{(x+5)^7\sin x}
f
(
x
)
=
(
x
+
5
)
7
sin
x
x
3/2
x
+
1
Find the slope of the tangent line to
g
(
x
)
=
x
x
2
+
sin
x
g(x)= x^x\sqrt{2+\sin x}
g
(
x
)
=
x
x
2
+
sin
x
when
x
=
π
x=\pi
x
=
π
.
Differentiate the following functions.
(a)
g
(
t
)
=
(
t
5
+
t
2
)
6
t
g(t)=(t^5+t^2)^{6t}
g
(
t
)
=
(
t
5
+
t
2
)
6
t
(b)
f
(
x
)
=
x
6
x
f(x)=\sqrt x^{6x}
f
(
x
)
=
x
6
x
(c)
f
(
x
)
=
(
x
3
+
4
)
tan
x
f(x)=(x^3+4)^{\tan x}
f
(
x
)
=
(
x
3
+
4
)
t
a
n
x
(d) Find
d
y
d
x
\frac{dy}{dx}
d
x
d
y
if
y
cos
(
x
)
=
x
tan
(
y
)
y\cos(x)=x\tan(y)
y
cos
(
x
)
=
x
tan
(
y
)
Calculate the derivative of the following function.
f
(
x
)
=
(
sin
x
)
cos
x
\displaystyle f(x)=(\sin{x})^{\cos{x}}
f
(
x
)
=
(
sin
x
)
c
o
s
x
Logarithmic Differentiation
Find
y
′
y'
y
′
for the function
𝑦
=
𝑥
2
−
x
2
𝑦 = 𝑥^{2-x^2}
y
=
x
2
−
x
2
.
Logarithmic Differentiation
Use logarithmic differentiation to find
g
′
g'
g
′
where
g
(
x
)
=
x
x
2
2
+
tan
x
g(x)=x^{x^2}\sqrt{2+\tan x}
g
(
x
)
=
x
x
2
2
+
tan
x
Compute the derivative of
f
(
x
)
=
x
3
/
2
x
+
1
(
x
+
5
)
7
sin
x
f(x) = \dfrac{x^{3/2}\sqrt{x+1}}{(x+5)^7\sin x}
f
(
x
)
=
(
x
+
5
)
7
sin
x
x
3/2
x
+
1
Find the derivative of
y
=
(
cos
x
)
5
x
2
y=\left(\cos x\right)^{\frac{5x}{2}}
y
=
(
cos
x
)
2
5
x
Logarithmic Differentiation: Big Product/Quotient
Consider
y
=
(
x
2
+
1
)
2
(
x
−
1
)
4
(
x
3
+
2
)
3
3
\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}
y
=
3
(
x
3
+
2
)
3
(
x
2
+
1
)
2
(
x
−
1
)
4
. Compute
d
y
d
x
at
x
=
0
\displaystyle\frac{dy}{dx}\text{ at }x=0
d
x
d
y
at
x
=
0
.
Logarithmic for Implicitly Defined Function
Find
d
y
/
d
x
dy/dx
d
y
/
d
x
at the point
(
1
,
1
)
(1,1)
(
1
,
1
)
given
x
y
=
y
x
x^y=y^x
x
y
=
y
x
Logarithmic with Chain
Use logarithmic differentiation to find
g
′
g^\prime
g
′
where
g
(
x
)
=
x
x
2
+
sin
x
\displaystyle g(x)=x^x\sqrt{2+\sin{x}}
g
(
x
)
=
x
x
2
+
sin
x
.
Differentiation: Logarithmic with Trig
If
y
=
sin
x
x
y=\sin^xx
y
=
sin
x
x
, find
y
′
y'
y
′
.
Logarithmic Differentiation
Consider
y
=
(
x
2
+
1
)
2
(
x
−
1
)
4
(
x
3
+
2
)
3
3
\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}
y
=
3
(
x
3
+
2
)
3
(
x
2
+
1
)
2
(
x
−
1
)
4
. Compute
d
y
d
x
at
x
=
0
\displaystyle\frac{dy}{dx}\text{ at }x=0
d
x
d
y
at
x
=
0
.
Logarithmic differentiation
Find the derivative of
f
(
x
)
=
(
x
+
1
)
9
e
x
(
sin
x
)
3
2
x
5
x
2
+
1
f\left(x\right)=\frac{\left(x+1\right)^9e^x\left(\sin\ x\right)^{\frac{3}{2}}}{x^5\sqrt{x^2+1}}
f
(
x
)
=
x
5
x
2
+
1
(
x
+
1
)
9
e
x
(
s
i
n
x
)
2
3
.
Logarithmic Differentiation
Find the derivative of
y
=
(
x
−
3
)
9
x
x
ln
x
x
−
9
.
y=\frac{\left(x-3\right)^9\ x^x\ \ln\ x}{x-9}.
y
=
x
−
9
(
x
−
3
)
9
x
x
l
n
x
.
Logarithmic Differentiation
If
f
(
x
)
=
(
2
x
)
sin
x
f(x)=(2x)^{\sin x}
f
(
x
)
=
(
2
x
)
s
i
n
x
, find
f
′
(
π
2
)
f'(\frac{\pi}{2})
f
′
(
2
π
)
?
Logarithmic Differentiation
If
f
(
x
)
=
x
x
f(x)=x^{\sqrt{x}}
f
(
x
)
=
x
x
then find
f
′
(
4
)
f'(4)
f
′
(
4
)
.
Find the derivative of
f
(
x
)
=
(
tan
x
)
ln
x
.
f(x)=(\tan x)^{\ln\ x}.
f
(
x
)
=
(
tan
x
)
l
n
x
.
Find the derivative of
f
(
x
)
=
(
tan
x
)
ln
x
f(x)=(\tan x)^{\ln x}
f
(
x
)
=
(
tan
x
)
l
n
x
.
Calculate the derivative of the following functions.
f
(
x
)
=
(
tan
x
)
sec
x
\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
f
(
x
)
=
(
tan
x
)
sec
x
Find
f
′
(
1
)
f'(1)
f
′
(
1
)
if
f
(
x
)
=
(
arctan
x
)
x
2
f\left(x\right)=\left(\arctan x\right)^{x^2}
f
(
x
)
=
(
arctan
x
)
x
2
.
Practice: Log Differentiation
Practice: Log Differentiation
Find the derivative of
x
2
(
cos
x
)
(
e
2
x
)
2
x
ln
(
x
)
\frac{x^2\left(\cos x\right)\left(e^{2x}\right)}{2x\ln\left(x\right)}
2
x
l
n
(
x
)
x
2
(
c
o
s
x
)
(
e
2
x
)
Complicated Log Derivative
If
f
(
x
)
=
(
sin
x
)
3
x
f\left(x\right)=\left(\sin x\right)^{3x}
f
(
x
)
=
(
sin
x
)
3
x
, then
f
′
(
x
)
=
f'\left(x\right)=
f
′
(
x
)
=
Logarithmic differentiation
Find the derivative of the function
y
=
sin
(
x
)
sin
(
x
)
y = \sin(x)^{\sin(x)}
y
=
sin
(
x
)
s
i
n
(
x
)
Logarithmic Differentiation
Calculate the derivative of the following functions.
f
(
x
)
=
(
sin
x
)
cos
x
\displaystyle f(x)=(\sin{x})^{\cos{x}}
f
(
x
)
=
(
sin
x
)
c
o
s
x
Logarithmic Differentiation
Calculate the derivative of the following functions.
f
(
x
)
=
x
x
2
+
sin
x
\displaystyle f(x)=x^x\sqrt{2+\sin{x}}
f
(
x
)
=
x
x
2
+
sin
x
Logarithmic Differentiation
Calculate the derivative of the following functions.
f
(
x
)
=
(
tan
x
)
sec
x
\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
f
(
x
)
=
(
tan
x
)
sec
x
Practice: Logarithmic Differentiation
Find the derivative of
f
(
x
)
=
sin
x
cos
x
f(x)=\sin{x}^{\cos{x}}
f
(
x
)
=
sin
x
c
o
s
x
More Derivatives of Inverse Trig Functions Questions:
Inverse trigonometric derivative
Find the derivative of
g
(
x
)
=
(
cos
−
1
x
)
(
sin
−
1
x
)
g(x)=\left(\cos^{-1}x\right)\left(\sin^{-1}x\right)
g
(
x
)
=
(
cos
−
1
x
)
(
sin
−
1
x
)
.
Derivatives: Inverse Trigonometric Functions
Given that
f
(
x
)
=
2
−
arcsin
(
x
)
f\left(x\right)=2-\arcsin\left(x\right)
f
(
x
)
=
2
−
arcsin
(
x
)
, find
f
′
(
0
)
f'\left(0\right)
f
′
(
0
)
Inverse Trigonometric Functions
Compute the derivative of
f
(
x
)
=
arcsin
(
x
+
2
)
f\left(x\right)=\arcsin\left(x+2\right)
f
(
x
)
=
arcsin
(
x
+
2
)
Derivatives: Inverse Trigonometric Functions
Given that
f
(
x
)
=
(
2
−
arcsin
(
x
2
)
)
3
f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3
f
(
x
)
=
(
2
−
arcsin
(
x
2
)
)
3
, find
f
′
(
0
)
f'\left(0\right)
f
′
(
0
)
The Chain Rule
Q
:
\bf{Q:}
Q
:
Find
(
3
3
x
−
x
4
+
2
cos
−
1
(
x
)
−
2025
)
′
(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
(
3
3
x
−
x
4
+
2
c
o
s
−
1
(
x
)
−
2025
)
′
Derivatives
Find the derivative of the following functions at the given values
a)
f
(
x
)
=
arcsin
(
x
2
)
x
3
−
1
f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}
f
(
x
)
=
x
3
−
1
a
r
c
s
i
n
(
x
2
)
at
x
=
0
x=0
x
=
0
b)
g
(
x
)
=
(
cos
x
)
e
x
g\left(x\right)=\left(\cos x\right)^{e^x}
g
(
x
)
=
(
cos
x
)
e
x
at
x
=
0
x=0
x
=
0
Finding a Derivative
If
f
(
x
)
=
tan
−
1
(
3
sin
x
)
f(x)=\tan^{-1}\left(3^{\sin x}\right)
f
(
x
)
=
tan
−
1
(
3
s
i
n
x
)
, find
f
′
(
π
)
f'(\pi)
f
′
(
π
)
.
The Chain Rule
Q
:
\bf{Q:}
Q
:
Find
(
3
3
x
−
x
4
+
2
c
o
s
−
1
(
x
)
−
2019
)
′
(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
(
3
3
x
−
x
4
+
2
co
s
−
1
(
x
)
−
2019
)
′
Inverse trigonometric derivative
Find the derivative of
g
(
x
)
=
cos
−
1
(
sin
−
1
x
)
g(x)=\cos^{-1}\left(\sin^{-1}x\right)
g
(
x
)
=
cos
−
1
(
sin
−
1
x
)
.
Find
d
d
x
(
arctan
(
e
tan
x
)
)
\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)
d
x
d
(
arctan
(
e
t
a
n
x
)
)
.
Inverse trigonometric derivative
Find the derivative of
g
(
x
)
=
cos
−
1
(
sin
−
1
x
)
g(x)=\cos^{-1}\left(\sin^{-1}x\right)
g
(
x
)
=
cos
−
1
(
sin
−
1
x
)
.
Inverse Trigonometric Functions
Compute the derivative of
f
(
x
)
=
arcsin
(
x
+
2
x
−
1
)
\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
f
(
x
)
=
arcsin
(
x
−
1
x
+
2
)
Derivatives: Inverse Trigonometric Functions
Given that
f
(
x
)
=
(
2
−
arcsin
(
x
2
)
)
3
f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3
f
(
x
)
=
(
2
−
arcsin
(
x
2
)
)
3
, find
f
′
(
0
)
f'\left(0\right)
f
′
(
0
)
.
Evaluating a derivative at a point
Evaluate
(
cos
−
1
(
x
2
)
)
′
∣
0
\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
(
cos
−
1
(
x
2
)
)
′
∣
0
Inverse Trig Derivative
Find the equation of the line tangent to
f
(
x
)
=
sin
−
1
x
1
+
x
f(x)=\frac{\sin^{-1}x}{1+x}
f
(
x
)
=
1
+
x
sin
−
1
x
at
x
=
1
/
2
x=1/2
x
=
1/2
.
Inverse Trigonometric Functions
Compute the derivative of
f
(
x
)
=
arcsin
(
x
+
2
x
−
1
)
\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
f
(
x
)
=
arcsin
(
x
−
1
x
+
2
)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that
f
(
x
)
=
(
2
−
arcsin
(
x
2
)
)
3
f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3
f
(
x
)
=
(
2
−
arcsin
(
x
2
)
)
3
, find
f
′
(
0
)
f'\left(0\right)
f
′
(
0
)
.
Evaluating a derivative at a point
Evaluate
(
cos
−
1
(
x
2
)
)
′
∣
0
\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
(
cos
−
1
(
x
2
)
)
′
∣
0
Inverse Trig Derivative
Find the equation of the line tangent to
f
(
x
)
=
sin
−
1
x
1
+
x
f(x)=\frac{\sin^{-1}x}{1+x}
f
(
x
)
=
1
+
x
sin
−
1
x
at
x
=
1
/
2
x=1/2
x
=
1/2
.
Inverse trigonometric derivative
Find the derivative of
g
(
x
)
=
cos
−
1
(
sin
−
1
x
)
g(x)=\cos^{-1}\left(\sin^{-1}x\right)
g
(
x
)
=
cos
−
1
(
sin
−
1
x
)
.
Practice: Chain Rule
Find the derivative of
y
=
ln
(
arctan
x
)
y=\ln(\arctan x)
y
=
ln
(
arctan
x
)
Find
d
d
x
(
arctan
(
e
tan
x
)
)
\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)
d
x
d
(
arctan
(
e
t
a
n
x
)
)
.
Evaluating a derivative at a point
Evaluate
(
cos
−
1
(
x
2
)
)
′
∣
0
\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
(
cos
−
1
(
x
2
)
)
′
∣
0
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a)
f
(
x
)
=
arcsin
(
3
x
2
)
f(x)=\arcsin(3x^2)
f
(
x
)
=
arcsin
(
3
x
2
)
The Chain Rule
Q
:
\bf{Q:}
Q
:
Find
(
3
3
x
−
x
4
+
2
c
o
s
−
1
(
x
)
−
2019
)
′
(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
(
3
3
x
−
x
4
+
2
co
s
−
1
(
x
)
−
2019
)
′
Derivatives: Inverse Trigonometric Functions
Find
[
tan
−
1
(
x
3
)
]
′
[\tan^{-1}(\sqrt[3]x)]'
[
tan
−
1
(
3
x
)
]
′
Derivatives: Inverse Trigonometric Functions
Show that
d
d
x
sin
−
1
x
=
1
1
−
x
2
\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
d
x
d
sin
−
1
x
=
1
−
x
2
1
Practice: Derivative of Inverse Trig
Q:
\textbf{Q:}
Q:
Show that
d
d
x
cos
−
1
x
=
−
1
1
−
x
2
.
\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
d
x
d
cos
−
1
x
=
−
1
−
x
2
1
.
Practice: Tangent Line with Inverse Trig
Q:
\textbf{Q:}
Q:
Find the equation of the tangent line to the graph of
f
(
x
)
=
cot
−
1
(
x
2
)
\displaystyle f(x)=\cot^{-1}(x^{2})
f
(
x
)
=
cot
−
1
(
x
2
)
at the point
x
=
2
x=2
x
=
2
.
Inverse Trig with Log
Compute the derivative of
f
(
x
)
=
arctan
(
log
10
x
)
\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
f
(
x
)
=
arctan
(
log
10
x
)
Inverse Trigonometric Functions
Compute the derivative of
f
(
x
)
=
arcsin
(
x
+
2
x
−
1
)
\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
f
(
x
)
=
arcsin
(
x
−
1
x
+
2
)
Derivatives: Exponential and Trigonometric Functions
Find the derivative of
f
(
x
)
=
5
x
6
−
sec
−
1
(
e
2
x
)
+
cos
−
1
x
f(x)=5x^6-\sec^{-1}(e^{2x})+\cos^{-1}x
f
(
x
)
=
5
x
6
−
sec
−
1
(
e
2
x
)
+
cos
−
1
x
Inverse Trig Derivative
Find the equation of the line tangent to
f
(
x
)
=
sin
−
1
x
1
+
x
f(x)=\frac{\sin^{-1}x}{1+x}
f
(
x
)
=
1
+
x
sin
−
1
x
at
x
=
1
/
2
x=1/2
x
=
1/2
.
Derivatives: Inverse Trigonometric Functions
Find
f
′
(
x
)
f'(x)
f
′
(
x
)
if
f
(
x
)
=
arctan
x
f(x)=\arctan \sqrt{x}
f
(
x
)
=
arctan
x
Differentiate the following
y
=
tan
−
1
x
2
−
1
y=\tan^{-1}\sqrt{x^2-1}
y
=
tan
−
1
x
2
−
1
Find the derivative of the following function.
f
(
x
)
=
arcsinx
\displaystyle f(x)=\sqrt{\text{arcsin{x}}}
f
(
x
)
=
arcsin
x
The Chain Rule
Q
:
\bf{Q:}
Q
:
Find
(
3
3
x
−
x
4
+
2
c
o
s
−
1
(
x
)
−
2019
)
′
(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
(
3
3
x
−
x
4
+
2
co
s
−
1
(
x
)
−
2019
)
′
Derivatives: Inverse Trigonometric Functions
Find
[
tan
−
1
(
x
3
)
]
′
[\tan^{-1}(\sqrt[3]x)]'
[
tan
−
1
(
3
x
)
]
′
Derivatives: Inverse Trigonometric Functions
Show that
d
d
x
sin
−
1
x
=
1
1
−
x
2
\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
d
x
d
sin
−
1
x
=
1
−
x
2
1
Practice: Derivative of Inverse Trig
Q:
\textbf{Q:}
Q:
Show that
d
d
x
cos
−
1
x
=
−
1
1
−
x
2
.
\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
d
x
d
cos
−
1
x
=
−
1
−
x
2
1
.
Practice: Tangent Line with Inverse Trig
Q:
\textbf{Q:}
Q:
Find the equation of the tangent line to the graph of
f
(
x
)
=
cot
−
1
(
x
2
)
\displaystyle f(x)=\cot^{-1}(x^{2})
f
(
x
)
=
cot
−
1
(
x
2
)
at the point
x
=
2
x=2
x
=
2
.
Inverse Trig with Log
Compute the derivative of
f
(
x
)
=
arctan
(
log
10
x
)
\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
f
(
x
)
=
arctan
(
log
10
x
)
Inverse Trigonometric Functions
Compute the derivative of
f
(
x
)
=
arcsin
(
x
+
2
x
−
1
)
\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
f
(
x
)
=
arcsin
(
x
−
1
x
+
2
)
Derivatives: Exponential Functions, Inverse Trigonometric Functions
Compute the derivative of
f
(
x
)
=
sec
(
3
2
x
)
f(x)=\sec(3^{2x})
f
(
x
)
=
sec
(
3
2
x
)
.
Derivatives: Logarithmic and Inverse Trigonometric Functions
Evaluate
d
d
x
(
sin
−
1
x
sin
x
log
3
x
)
\displaystyle \frac{\text{d}}{\text{d}x}\left( \frac{\sin^{-1}x\sin x}{\log_3x}\right)
d
x
d
(
lo
g
3
x
sin
−
1
x
sin
x
)
.
Find the derivative of
f
(
x
)
=
arcsin
(
e
x
+
1
)
f(x)=\arcsin(e^{x+1})
f
(
x
)
=
arcsin
(
e
x
+
1
)
.
Inverse trigonometric derivative
Find the derivative of
g
(
x
)
=
cos
−
1
(
sin
−
1
x
)
g(x)=\cos^{-1}\left(\sin^{-1}x\right)
g
(
x
)
=
cos
−
1
(
sin
−
1
x
)
.
Derivatives: Inverse Trigonometric Functions
Find the derivative of
f
(
x
)
=
(
arccos
x
)
2
sec
x
f\left(x\right)=\frac{\left(\arccos x\right)^2}{\sec x}
f
(
x
)
=
s
e
c
x
(
a
r
c
c
o
s
x
)
2
.
Evaluating a derivative at a point
Evaluate
(
cos
−
1
(
x
2
)
)
′
∣
0
\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
(
cos
−
1
(
x
2
)
)
′
∣
0
Practice: Chain Rule
Find the derivative of
y
=
ln
(
arctan
x
)
y=\ln(\arctan x)
y
=
ln
(
arctan
x
)
Find
d
d
x
(
arctan
(
e
tan
x
)
)
\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)
d
x
d
(
arctan
(
e
t
a
n
x
)
)
.
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a)
f
(
x
)
=
arcsin
(
3
x
2
)
f(x)=\arcsin(3x^2)
f
(
x
)
=
arcsin
(
3
x
2
)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that
f
(
x
)
=
(
2
−
arcsin
(
x
2
)
)
3
f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3
f
(
x
)
=
(
2
−
arcsin
(
x
2
)
)
3
, find
f
′
(
0
)
f'\left(0\right)
f
′
(
0
)
.
Finding a Derivative
Practice: Finding a Derivative
(
tan
−
1
(
3
sin
x
)
)
′
∣
x
=
π
=
\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
(
tan
−
1
(
3
s
i
n
x
)
)
′
∣
x
=
π
=
Linear Approximation and Trigonometric Derivative
a) Show that for
f
(
x
)
=
arcsin
(
x
)
f(x) = \arcsin(x)
f
(
x
)
=
arcsin
(
x
)
, that
f
′
(
x
)
=
1
1
−
x
2
f'(x) = \frac{1}{\sqrt{1 - x^2}}
f
′
(
x
)
=
1
−
x
2
1
.
b) Use linear approximation at a suitable close value to estimate
arcsin
(
0.1
)
\arcsin(0.1)
arcsin
(
0.1
)
. Your solution may be left in terms of fractions.
Derivatives: Inverse Trigonometric Functions
Find the derivative of the following function.
f
(
x
)
=
1
arcsin
(
x
)
+
arcsin
(
1
x
)
\displaystyle f(x)=\frac{1}{\text{arcsin}(x)}+\text{arcsin}\left(\frac{1}{x}\right)
f
(
x
)
=
arcsin
(
x
)
1
+
arcsin
(
x
1
)
Inverse Trigonometric Functions
Calculate the derivative of the following functions.
f
(
x
)
=
arcsin
(
e
x
+
1
)
\displaystyle f(x)=\text{arcsin}(e^{x+1})
f
(
x
)
=
arcsin
(
e
x
+
1
)
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f
(
x
)
=
tan
(
arccos
(
e
4
x
)
)
\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))
f
(
x
)
=
tan
(
arccos
(
e
4
x
))
More The Chain Rule Questions:
The Chain Rule
Q
:
\bf{Q:}
Q
:
Find
(
3
3
x
−
x
4
+
2
cos
−
1
(
x
)
−
2025
)
′
(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
(
3
3
x
−
x
4
+
2
c
o
s
−
1
(
x
)
−
2025
)
′
Derivatives
Find the derivative of the following functions at the given values
a)
f
(
x
)
=
arcsin
(
x
2
)
x
3
−
1
f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}
f
(
x
)
=
x
3
−
1
a
r
c
s
i
n
(
x
2
)
at
x
=
0
x=0
x
=
0
b)
g
(
x
)
=
(
cos
x
)
e
x
g\left(x\right)=\left(\cos x\right)^{e^x}
g
(
x
)
=
(
cos
x
)
e
x
at
x
=
0
x=0
x
=
0
Finding a Derivative
If
f
(
x
)
=
tan
−
1
(
3
sin
x
)
f(x)=\tan^{-1}\left(3^{\sin x}\right)
f
(
x
)
=
tan
−
1
(
3
s
i
n
x
)
, find
f
′
(
π
)
f'(\pi)
f
′
(
π
)
.
The Chain Rule
Q
:
\bf{Q:}
Q
:
Find
(
3
3
x
−
x
4
+
2
c
o
s
−
1
(
x
)
−
2019
)
′
(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
(
3
3
x
−
x
4
+
2
co
s
−
1
(
x
)
−
2019
)
′
The Chain Rule
Find the derivative of
h
(
x
)
=
f
(
g
(
x
)
)
h(x)=f(g(x))
h
(
x
)
=
f
(
g
(
x
))
if
f
(
x
)
=
x
2
+
1
f(x)=x^2+1
f
(
x
)
=
x
2
+
1
and
g
(
x
)
=
3
x
g(x)=3\sqrt{x}
g
(
x
)
=
3
x
.
The Chain Rule
Find the derivative of
h
(
x
)
=
f
(
g
(
x
)
)
h(x)=f(g(x))
h
(
x
)
=
f
(
g
(
x
))
if
f
(
x
)
=
x
2
+
1
f(x)=x^2+1
f
(
x
)
=
x
2
+
1
and
g
(
x
)
=
3
x
g(x)=3\sqrt{x}
g
(
x
)
=
3
x
.
The Chain Rule
Find
g
′
(
0
)
g'(0)
g
′
(
0
)
for
g
(
x
)
=
f
(
x
2
)
+
f
(
x
)
,
g(x)=\sqrt{f(x^2)} + f(x),
g
(
x
)
=
f
(
x
2
)
+
f
(
x
)
,
given
f
(
0
)
=
2
and
f
′
(
0
)
=
1.
f(0)=2 \text{ and }f'(0)=1.
f
(
0
)
=
2
and
f
′
(
0
)
=
1.
The Chain Rule
If
2
−
g
(
x
)
=
x
2
+
2
[
f
(
x
)
]
2
−
x
3
g
(
x
)
2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right)
2
−
g
(
x
)
=
x
2
+
2
[
f
(
x
)
]
2
−
x
3
g
(
x
)
,
f
(
0
)
=
1
,
f
′
(
0
)
=
−
2
,
g
(
0
)
=
−
1
and
g
′
′
(
0
)
=
3
f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3
f
(
0
)
=
1
,
f
′
(
0
)
=
−
2
,
g
(
0
)
=
−
1
and
g
′′
(
0
)
=
3
, then the value of
f
′
′
(
0
)
f''\left(0\right)
f
′′
(
0
)
is equal to
Find
g
′
(
0
)
g^{\prime}(0)
g
′
(
0
)
for
g
(
x
)
=
f
(
x
2
)
+
f
(
x
)
g(x)=\sqrt{f(x^2)}+f(x)
g
(
x
)
=
f
(
x
2
)
+
f
(
x
)
, given
f
(
0
)
=
2
f(0)=2
f
(
0
)
=
2
and
f
′
(
0
)
=
1
f^{\prime}(0)=1
f
′
(
0
)
=
1
.
The Chain Rule
Find the derivative of the following function.
f
(
x
)
=
(
2
x
+
1
x
2
+
1
)
3
\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
f
(
x
)
=
(
x
2
+
1
2
x
+
1
)
3
Find
d
d
x
(
arctan
(
e
tan
x
)
)
\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)
d
x
d
(
arctan
(
e
t
a
n
x
)
)
.
The Chain Rule
The derivative of
y
=
3
e
x
y=3^{e^x}
y
=
3
e
x
is
The Chain Rule
The derivative of
y
=
3
e
x
y=3^{e^x}
y
=
3
e
x
is
The Chain Rule
The derivative of
y
=
3
e
x
y=3^{e^x}
y
=
3
e
x
is
The Chain Rule
The derivative of
y
=
3
e
x
y=3^{e^x}
y
=
3
e
x
is
Practice: Chain Rule
Find the derivative of
y
=
ln
(
arctan
x
)
y=\ln(\arctan x)
y
=
ln
(
arctan
x
)
Find
d
d
x
(
arctan
(
e
tan
x
)
)
\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)
d
x
d
(
arctan
(
e
t
a
n
x
)
)
.
The Chain Rule: Finding a derivative
Find the derivative of
f
(
x
)
=
cos
(
e
2
x
)
f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
f
(
x
)
=
cos
(
e
2
x
)
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a)
f
(
x
)
=
arcsin
(
3
x
2
)
f(x)=\arcsin(3x^2)
f
(
x
)
=
arcsin
(
3
x
2
)
The Chain Rule
Given this table of values for
f
,
g
,
f
′
,
and
g
′
f,\ g,\ f',\ \text{and}\ g'
f
,
g
,
f
′
,
and
g
′
below, answer the following questions.
x
f
(
x
)
f
′
(
x
)
f
′
′
(
x
)
g
(
x
)
g
′
(
x
)
0
0
−
1
−
5
2
3
π
2
π
1
0
4
5
2
−
2
−
4
10
π
2
−
3
\begin{array}{|c|c|c|c|c|c|} \hline x&f(x)&f'(x)&f''(x)&g(x)&g'(x)\\ \hline 0&0&-1&-5&2&3\\ \hline \frac{\pi}{2}&\pi&1&0&4&5\\ \hline 2&-2&-4&10&\frac{\pi}{2}&-3\\ \hline \end{array}
x
0
2
π
2
f
(
x
)
0
π
−
2
f
′
(
x
)
−
1
1
−
4
f
′′
(
x
)
−
5
0
10
g
(
x
)
2
4
2
π
g
′
(
x
)
3
5
−
3
The Chain Rule
Q
:
\bf{Q:}
Q
:
Find
(
3
3
x
−
x
4
+
2
c
o
s
−
1
(
x
)
−
2019
)
′
(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
(
3
3
x
−
x
4
+
2
co
s
−
1
(
x
)
−
2019
)
′
The Chain Rule
Find the derivative of
h
(
x
)
=
f
(
g
(
x
)
)
h(x)=f(g(x))
h
(
x
)
=
f
(
g
(
x
))
if
f
(
x
)
=
x
2
+
1
f(x)=x^2+1
f
(
x
)
=
x
2
+
1
and
g
(
x
)
=
3
x
g(x)=3\sqrt{x}
g
(
x
)
=
3
x
.
Practice: Chain Rule with Given Values
Q.
\textbf{Q.}
Q.
Given that
g
(
1
)
=
g
′
(
1
)
=
1
g(1)=g^{\prime}(1)=1
g
(
1
)
=
g
′
(
1
)
=
1
, find
f
′
(
π
/
2
)
f^{\prime}(\pi/2)
f
′
(
π
/2
)
if
f
(
x
)
=
g
(
sin
x
)
x
+
1
\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}
f
(
x
)
=
x
+
1
g
(
sin
x
)
.
Practice: Chain Rule with Given Values
Q:
\textbf{Q:}
Q:
Given that
f
(
1
)
=
5
,
g
(
1
)
=
−
2
,
f
′
(
1
)
=
2
f(1)=5,\ g(1)=-2,\ f'(1)=2
f
(
1
)
=
5
,
g
(
1
)
=
−
2
,
f
′
(
1
)
=
2
and
g
′
(
1
)
=
−
1
/
2
g'(1)=-1/2
g
′
(
1
)
=
−
1/2
, find the derivative of
f
(
x
)
−
g
2
(
x
)
\sqrt{f(x)-g^2(x)}
f
(
x
)
−
g
2
(
x
)
at
x
=
1
x=1
x
=
1
.
The Quotient and Chain Rules
Let
f
(
x
)
=
x
2
−
6
x
−
3
\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}
f
(
x
)
=
x
−
3
x
2
−
6
.
The chain rule
Find the derivative of
h
(
x
)
=
log
(
cos
(
x
)
)
h(x) = \log(\cos(x))
h
(
x
)
=
lo
g
(
cos
(
x
))
. Remember that
log
x
=
log
e
x
=
ln
x
\log x = \log_e x = \ln x
lo
g
x
=
lo
g
e
x
=
ln
x
.
The chain rule: Logarithmic Derivatives
The derivative of
log
3
(
e
3
x
)
\log_3\left(e^{3x}\right)
lo
g
3
(
e
3
x
)
is
The chain rule
The derivative of
y
=
(
2
−
x
7
)
500
y=\left(2-x^7\right)^{500}
y
=
(
2
−
x
7
)
500
is
The Chain Rule
If
2
−
g
(
x
)
=
x
2
+
2
[
f
(
x
)
]
2
−
x
3
g
(
x
)
2-g\left(x\right)=x^2+2\left[f\left(x\right)\right]^2-x^3g\left(x\right)
2
−
g
(
x
)
=
x
2
+
2
[
f
(
x
)
]
2
−
x
3
g
(
x
)
,
f
(
0
)
=
1
,
f
′
(
0
)
=
−
2
,
g
(
0
)
=
−
1
and
g
′
′
(
0
)
=
3
f\left(0\right)=1\ ,\ f'\left(0\right)=-2,\ g\left(0\right)=-1\ \text{and}\ \ g''\left(0\right)=3
f
(
0
)
=
1
,
f
′
(
0
)
=
−
2
,
g
(
0
)
=
−
1
and
g
′′
(
0
)
=
3
, then the value of
f
′
′
(
0
)
f''\left(0\right)
f
′′
(
0
)
is equal to
Find the derivative of
h
(
x
)
=
f
(
g
(
x
)
)
if
f
(
x
)
=
x
2
+
1
h(x)=f(g(x))\,\, \text{ if }\, f(x)=x^2+1
h
(
x
)
=
f
(
g
(
x
))
if
f
(
x
)
=
x
2
+
1
and
g
(
x
)
=
3
x
g(x)=3\sqrt{x}
g
(
x
)
=
3
x
.
Find
g
′
(
0
)
g^{\prime}(0)
g
′
(
0
)
for
g
(
x
)
=
f
(
x
2
)
+
f
(
x
)
g(x)=\sqrt{f(x^2)}+f(x)
g
(
x
)
=
f
(
x
2
)
+
f
(
x
)
, given
f
(
0
)
=
2
f(0)=2
f
(
0
)
=
2
and
f
′
(
0
)
=
1
f^{\prime}(0)=1
f
′
(
0
)
=
1
.
Find the derivative of
h
(
x
)
=
f
(
g
(
x
)
)
if
f
(
x
)
=
x
2
+
1
h(x)=f(g(x))\,\, \text{ if }\, f(x)=x^2+1
h
(
x
)
=
f
(
g
(
x
))
if
f
(
x
)
=
x
2
+
1
and
g
(
x
)
=
3
x
g(x)=3\sqrt{x}
g
(
x
)
=
3
x
.
Find the derivative of the following function.
f
(
x
)
=
(
2
x
+
1
x
2
+
1
)
3
\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
f
(
x
)
=
(
x
2
+
1
2
x
+
1
)
3
The Chain Rule
Find
g
′
(
0
)
g'(0)
g
′
(
0
)
for
g
(
x
)
=
f
(
x
2
)
+
f
(
x
)
,
g(x)=\sqrt{f(x^2)} + f(x),
g
(
x
)
=
f
(
x
2
)
+
f
(
x
)
,
given
f
(
0
)
=
2
and
f
′
(
0
)
=
1.
f(0)=2 \text{ and }f'(0)=1.
f
(
0
)
=
2
and
f
′
(
0
)
=
1.
The Chain Rule
Compute the derivative of
f
(
x
)
=
x
4
+
4
x
4
+
4
f(x) = \sqrt{x^4 + \frac{4}{x^4}+4}
f
(
x
)
=
x
4
+
x
4
4
+
4
Practice: Chain Rule
Find the derivative of the following function:
y
=
sin
(
x
)
y=\sin(\sqrt{x})
y
=
sin
(
x
)
Practice: Complicated Chain-Rule
Find an expression for the derivative of the following function:
f
(
x
)
=
cos
(
e
2
x
)
f\left(x\right)=\sqrt{\cos \left(e^{2x}\right)}
f
(
x
)
=
cos
(
e
2
x
)
Practice: Chain Rule
Find the derivative of the following function:
y
=
e
sin
(
x
)
y=e^{\sin(\sqrt{x})}
y
=
e
s
i
n
(
x
)
Find the derivative of the following function.
f
(
x
)
=
arcsinx
\displaystyle f(x)=\sqrt{\text{arcsin{x}}}
f
(
x
)
=
arcsin
x
The Chain Rule
Q
:
\bf{Q:}
Q
:
Find
(
3
3
x
−
x
4
+
2
c
o
s
−
1
(
x
)
−
2019
)
′
(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
(
3
3
x
−
x
4
+
2
co
s
−
1
(
x
)
−
2019
)
′
The Chain Rule
Find the derivative of
h
(
x
)
=
f
(
g
(
x
)
)
h(x)=f(g(x))
h
(
x
)
=
f
(
g
(
x
))
if
f
(
x
)
=
x
2
+
1
f(x)=x^2+1
f
(
x
)
=
x
2
+
1
and
g
(
x
)
=
3
x
g(x)=3\sqrt{x}
g
(
x
)
=
3
x
.
Practice: Chain Rule with Given Values
Q.
\textbf{Q.}
Q.
Given that
g
(
1
)
=
g
′
(
1
)
=
1
g(1)=g^{\prime}(1)=1
g
(
1
)
=
g
′
(
1
)
=
1
, find
f
′
(
π
/
2
)
f^{\prime}(\pi/2)
f
′
(
π
/2
)
if
f
(
x
)
=
g
(
sin
x
)
x
+
1
\displaystyle f(x)=\frac{\sqrt{g(\sin{x})}}{x+1}
f
(
x
)
=
x
+
1
g
(
sin
x
)
.
Practice: Chain Rule with Given Values
Q:
\textbf{Q:}
Q:
Given that
f
(
1
)
=
5
,
g
(
1
)
=
−
2
,
f
′
(
1
)
=
2
f(1)=5,\ g(1)=-2,\ f'(1)=2
f
(
1
)
=
5
,
g
(
1
)
=
−
2
,
f
′
(
1
)
=
2
and
g
′
(
1
)
=
−
1
/
2
g'(1)=-1/2
g
′
(
1
)
=
−
1/2
, find the derivative of
f
(
x
)
−
g
2
(
x
)
\sqrt{f(x)-g^2(x)}
f
(
x
)
−
g
2
(
x
)
at
x
=
1
x=1
x
=
1
.
Given the information
g
(
x
)
=
1
+
[
f
(
x
)
]
2
such that
f
(
1
)
=
2
,
f
′
(
1
)
=
−
3
g(x)=\sqrt{1+[f(x)]^2} \ \text{ such that } f(1)=2,f'(1)=-3
g
(
x
)
=
1
+
[
f
(
x
)
]
2
such that
f
(
1
)
=
2
,
f
′
(
1
)
=
−
3
. Find
g
′
(
1
)
.
g'(1).
g
′
(
1
)
.
Express your answer as a fraction in lowest terms. If the answer is negative, put the negative sign in front of the entire fraction.
The Chain Rule" Derivatives of Trigonometric Functions
Calculate the derivative of the function
f
(
x
)
=
tan
(
x
3
)
f\left(x\right)=\sqrt{\tan\left(x^3\right)}
f
(
x
)
=
tan
(
x
3
)
The Chain Rule
If
g
(
t
)
=
9
+
f
(
t
)
3
g\left(t\right)=\sqrt[3]{9+f(t)}
g
(
t
)
=
3
9
+
f
(
t
)
, write an expression for the derivative
g
′
(
t
)
g'\left(t\right)
g
′
(
t
)
The Chain Rule
Find the derivative of the following function.
f
(
x
)
=
(
2
x
+
1
x
2
+
1
)
3
\displaystyle f(x)=\left(\frac{2x+1}{x^2+1}\right)^3
f
(
x
)
=
(
x
2
+
1
2
x
+
1
)
3
Practice: Chain Rule
Find the derivative of
y
=
ln
(
arctan
x
)
y=\ln(\arctan x)
y
=
ln
(
arctan
x
)
Find
d
d
x
(
arctan
(
e
tan
x
)
)
\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right)
d
x
d
(
arctan
(
e
t
a
n
x
)
)
.
Practice: trig and inverse trig derivatives
Practice: trig and inverse trig derivatives
Find the derivative of the following trigonometric and inverse trigonometric functions:
a)
f
(
x
)
=
arcsin
(
3
x
2
)
f(x)=\arcsin(3x^2)
f
(
x
)
=
arcsin
(
3
x
2
)
The Chain Rule: Finding a derivative
Find the derivative of
f
(
x
)
=
cos
(
e
2
x
)
f\left(x\right)=\sqrt{\cos\left(e^{2x}\right)}
f
(
x
)
=
cos
(
e
2
x
)
(Duplicated)
Find
f
′
(
1
)
f'\left(1\right)
f
′
(
1
)
if
f
(
x
)
=
x
2
g
(
x
)
f\left(x\right)=\sqrt{x^2\ g\left(x\right)}
f
(
x
)
=
x
2
g
(
x
)
,
g
(
1
)
=
1
g\left(1\right)=1
g
(
1
)
=
1
and
g
′
(
1
)
=
2
g'\left(1\right)=2
g
′
(
1
)
=
2
.
(Duplicated)
If
f
(
x
)
=
(
g
(
x
)
−
h
(
x
)
)
5
f\left(x\right)=\left(g\left(x\right)-h\left(x\right)\right)^5
f
(
x
)
=
(
g
(
x
)
−
h
(
x
)
)
5
,
g
(
0
)
=
2
,
h
(
0
)
=
0
,
g
′
(
0
)
=
6
g\left(0\right)=2,\ h\left(0\right)=0,\ g'\left(0\right)=6
g
(
0
)
=
2
,
h
(
0
)
=
0
,
g
′
(
0
)
=
6
, and
h
′
(
0
)
=
−
4
h'\left(0\right)=-4
h
′
(
0
)
=
−
4
, find
f
′
(
0
)
f'\left(0\right)
f
′
(
0
)
.
The Chain Rule
The derivative of
y
=
3
e
x
y=3^{e^x}
y
=
3
e
x
is
The Chain Rule
Given this table of values for
f
,
g
,
f
′
,
and
g
′
f,\ g,\ f',\ \text{and}\ g'
f
,
g
,
f
′
,
and
g
′
below, answer the following questions.
x
f
(
x
)
f
′
(
x
)
f
′
′
(
x
)
g
(
x
)
g
′
(
x
)
0
0
−
1
−
5
2
3
π
2
π
1
0
4
5
2
−
2
−
4
10
π
2
−
3
\begin{array}{|c|c|c|c|c|c|} \hline x&f(x)&f'(x)&f''(x)&g(x)&g'(x)\\ \hline 0&0&-1&-5&2&3\\ \hline \frac{\pi}{2}&\pi&1&0&4&5\\ \hline 2&-2&-4&10&\frac{\pi}{2}&-3\\ \hline \end{array}
x
0
2
π
2
f
(
x
)
0
π
−
2
f
′
(
x
)
−
1
1
−
4
f
′′
(
x
)
−
5
0
10
g
(
x
)
2
4
2
π
g
′
(
x
)
3
5
−
3
Finding a Derivative
Practice: Finding a Derivative
(
tan
−
1
(
3
sin
x
)
)
′
∣
x
=
π
=
\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
(
tan
−
1
(
3
s
i
n
x
)
)
′
∣
x
=
π
=
Practice: The Chain Rule
Find the derivative of the following function
y
=
e
sin
x
y=e^{\sin\sqrt{x}}
y
=
e
s
i
n
x
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f
(
x
)
=
tan
(
arccos
(
e
4
x
)
)
\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))
f
(
x
)
=
tan
(
arccos
(
e
4
x
))
More The Quotient Rule Questions:
Derivatives
Find the derivative of the following functions at the given values
a)
f
(
x
)
=
arcsin
(
x
2
)
x
3
−
1
f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1}
f
(
x
)
=
x
3
−
1
a
r
c
s
i
n
(
x
2
)
at
x
=
0
x=0
x
=
0
b)
g
(
x
)
=
(
cos
x
)
e
x
g\left(x\right)=\left(\cos x\right)^{e^x}
g
(
x
)
=
(
cos
x
)
e
x
at
x
=
0
x=0
x
=
0
The quotient rule
Is
u
′
(
t
)
=
2
(
t
−
1
)
3
(
−
3
t
+
23
)
(
3
t
+
7
)
7
\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7}
u
′
(
t
)
=
(
3
t
+
7
)
7
2
(
t
−
1
)
3
(
−
3
t
+
23
)
the derivative of
u
(
t
)
=
(
t
−
1
)
4
(
3
t
+
7
)
6
\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6}
u
(
t
)
=
(
3
t
+
7
)
6
(
t
−
1
)
4
?
The quotient rule
Find the derivative of the following function:
g
(
x
)
=
x
+
1
x
+
1
\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}
g
(
x
)
=
x
+
1
x
+
1
The quotient rule
Is
u
′
(
t
)
=
2
(
t
−
1
)
3
(
−
3
t
+
23
)
(
3
t
+
7
)
7
\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7}
u
′
(
t
)
=
(
3
t
+
7
)
7
2
(
t
−
1
)
3
(
−
3
t
+
23
)
the derivative of
u
(
t
)
=
(
t
−
1
)
4
(
3
t
+
7
)
6
\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6}
u
(
t
)
=
(
3
t
+
7
)
6
(
t
−
1
)
4
?
The quotient rule
Given the function
f
(
x
)
=
g
(
x
)
h
(
x
)
−
e
x
cos
x
f\left(x\right)=\dfrac{g\left(x\right)}{h\left(x\right)}-e^x\cos x
f
(
x
)
=
h
(
x
)
g
(
x
)
−
e
x
cos
x
, and given
g
(
π
2
)
=
2
g\left(\dfrac{\pi}{2}\right)=2
g
(
2
π
)
=
2
,
g
′
(
π
2
)
=
3
g'\left(\dfrac{\pi}{2}\right)=3
g
′
(
2
π
)
=
3
,
h
(
π
2
)
=
1
h\left(\dfrac{\pi}{2}\right)=1
h
(
2
π
)
=
1
, and
h
′
(
π
2
)
=
2
h'\left(\dfrac{\pi}{2}\right)=2
h
′
(
2
π
)
=
2
, find
f
′
(
π
2
)
f'\left(\dfrac{\pi}{2}\right)
f
′
(
2
π
)
.
Derivatives: Exponential and Logarithmic Functions
Find
f
′
(
x
)
f'(x)
f
′
(
x
)
if
f
(
x
)
=
ln
x
x
e
x
\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}
f
(
x
)
=
x
e
x
ln
x
. Simplify.
Quotient and Product
Find the derivative of
g
(
t
)
=
(
1
+
2
t
7
t
)
(
t
2
−
1
)
\displaystyle g(t)=\left(\frac{1+2t}{7t}\right)(t^{2}-1)
g
(
t
)
=
(
7
t
1
+
2
t
)
(
t
2
−
1
)
Practice: Quotient Rule
Q.
\textbf{Q.}
Q.
Find the derivative of
f
(
x
)
=
x
2
+
3
x
4
/
3
+
1
x
2
+
1
\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
+
3
x
4/3
+
1
Find the derivative of the following function:
g
(
x
)
=
x
+
1
x
+
1
\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}
g
(
x
)
=
x
+
1
x
+
1
The Quotient and Chain Rules
Let
f
(
x
)
=
x
2
−
6
x
−
3
\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}
f
(
x
)
=
x
−
3
x
2
−
6
.
The Quotient Rule
Find the derivative of
g
(
x
)
=
x
2
−
5
2
x
+
1
g(x) = \frac{x^2 - 5}{2x + 1}
g
(
x
)
=
2
x
+
1
x
2
−
5
The Quotient Rule
Compute the derivative of
f
(
x
)
=
x
2
+
3
x
4
/
3
+
1
x
2
+
1
f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
+
3
x
4/3
+
1
Practice: Product Rule*
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
1
+
x
g
(
x
)
x
2
f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}
f
(
x
)
=
x
2
1
+
x
g
(
x
)
,
f
′
(
1
)
=
2
f'\left(1\right)=2
f
′
(
1
)
=
2
,
g
(
1
)
=
1
g\left(1\right)=1
g
(
1
)
=
1
, what is
g
′
(
1
)
g'\left(1\right)
g
′
(
1
)
?
The quotient rule
Given the function
f
(
x
)
=
g
(
x
)
h
(
x
)
−
e
x
cos
x
f\left(x\right)=\dfrac{g\left(x\right)}{h\left(x\right)}-e^x\cos x
f
(
x
)
=
h
(
x
)
g
(
x
)
−
e
x
cos
x
, and given
g
(
π
2
)
=
2
g\left(\dfrac{\pi}{2}\right)=2
g
(
2
π
)
=
2
,
g
′
(
π
2
)
=
3
g'\left(\dfrac{\pi}{2}\right)=3
g
′
(
2
π
)
=
3
,
h
(
π
2
)
=
1
h\left(\dfrac{\pi}{2}\right)=1
h
(
2
π
)
=
1
, and
h
′
(
π
2
)
=
2
h'\left(\dfrac{\pi}{2}\right)=2
h
′
(
2
π
)
=
2
, find
f
′
(
π
2
)
f'\left(\dfrac{\pi}{2}\right)
f
′
(
2
π
)
.
Derivatives: Exponential and Logarithmic Functions
Find
f
′
(
x
)
f'(x)
f
′
(
x
)
if
f
(
x
)
=
ln
x
x
e
x
\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}
f
(
x
)
=
x
e
x
ln
x
. Simplify.
The quotient rule
Is
u
′
(
t
)
=
2
(
t
−
1
)
3
(
−
3
t
+
23
)
(
3
t
+
7
)
7
\displaystyle u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7}
u
′
(
t
)
=
(
3
t
+
7
)
7
2
(
t
−
1
)
3
(
−
3
t
+
23
)
the derivative of
u
(
t
)
=
(
t
−
1
)
4
(
3
t
+
7
)
6
\displaystyle u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6}
u
(
t
)
=
(
3
t
+
7
)
6
(
t
−
1
)
4
?
Quotient and Product
Find the derivative of
g
(
t
)
=
(
1
+
2
t
7
t
)
(
t
2
−
1
)
\displaystyle g(t)=\left(\frac{1+2t}{7t}\right)(t^{2}-1)
g
(
t
)
=
(
7
t
1
+
2
t
)
(
t
2
−
1
)
Practice: Quotient Rule
Find the derivative of
f
(
x
)
=
x
2
+
3
x
4
/
3
+
1
x
2
+
1
\displaystyle f(x)=\frac{x^2+3x^{4/3}+1}{x^2+1}
f
(
x
)
=
x
2
+
1
x
2
+
3
x
4/3
+
1
The Quotient Rule
If
h
(
x
)
=
f
(
x
)
−
1
x
+
2
h\left(x\right)=\frac{f\left(x\right)-1}{x+2}
h
(
x
)
=
x
+
2
f
(
x
)
−
1
,
f
(
3
)
=
2
and
f
′
(
3
)
=
−
6
f\left(3\right)=2\ \text{and}\ f'\left(3\right)=-6
f
(
3
)
=
2
and
f
′
(
3
)
=
−
6
, calculate the value of
h
′
(
3
)
h'\left(3\right)
h
′
(
3
)
The Quotient Rule
Is
u
′
(
t
)
=
2
(
t
−
1
)
3
(
−
3
t
+
23
)
(
3
t
+
7
)
7
u'\left(t\right)=\frac{2\left(t-1\right)^3\left(-3t+23\right)}{\left(3t+7\right)^7}
u
′
(
t
)
=
(
3
t
+
7
)
7
2
(
t
−
1
)
3
(
−
3
t
+
23
)
the derivative of
u
(
t
)
=
(
t
−
1
)
4
(
3
t
+
7
)
6
u\left(t\right)=\frac{\left(t-1\right)^4}{\left(3t+7\right)^6}
u
(
t
)
=
(
3
t
+
7
)
6
(
t
−
1
)
4
?
Find the derivative with respect to
x
x
x
of the function
x
2030
3
+
x
2030
\displaystyle \frac{x^{2030}}{3+x^{2030}}
3
+
x
2030
x
2030
.
Compute the derivative of
f
(
x
)
=
x
2
+
3
x
4
/
3
+
1
x
2
+
1
f(x) =\dfrac{x^2 + 3x^{4/3} + 1}{x^2 + 1}
f
(
x
)
=
x
2
+
1
x
2
+
3
x
4/3
+
1
.
The Quotient Rule
Suppose
g
(
x
)
g\left(x\right)
g
(
x
)
is differentiable,
f
(
x
)
=
1
+
x
g
(
x
)
x
2
f\left(x\right)=\frac{1+x\ g\left(x\right)}{x^2}
f
(
x
)
=
x
2
1
+
x
g
(
x
)
,
f
′
(
1
)
=
2
f'\left(1\right)=2
f
′
(
1
)
=
2
,
g
(
1
)
=
1
g\left(1\right)=1
g
(
1
)
=
1
, what is
g
′
(
1
)
g'\left(1\right)
g
′
(
1
)
?
Find the derivative of the following function:
g
(
x
)
=
x
+
1
x
+
1
\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}
g
(
x
)
=
x
+
1
x
+
1
The quotient rule
Find the derivative of the following function:
g
(
x
)
=
x
+
1
x
+
1
\displaystyle g(x)=\frac{\sqrt{x+1}}{\sqrt{x}+1}
g
(
x
)
=
x
+
1
x
+
1