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Complicated Log Derivative
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Logarithmic Differentiation
3 Activities
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
)
=
(
cos
x
)
3
x
\left(\cos x\right)^{3x}
(
cos
x
)
3
x
3
(
sin
x
)
2
x
3\left(\sin x\right)^{2x}
3
(
sin
x
)
2
x
6
x
(
cos
x
)
2
x
6x\left(\cos x\right)^{2x}
6
x
(
cos
x
)
2
x
(
sin
x
)
3
x
[
3
ln
(
sin
x
)
+
3
x
⋅
cos
x
sin
x
]
\left(\sin x\right)^{3x}\left[3\ln\left(\sin x\right)+\frac{3x\cdot\cos x}{\sin x}\right]
(
sin
x
)
3
x
[
3
ln
(
sin
x
)
+
s
i
n
x
3
x
⋅
c
o
s
x
]
(
sin
x
)
3
x
[
cos
x
⋅
ln
(
sin
x
)
+
3
x
⋅
sin
x
]
\left(\sin x\right)^{3x}\left[\cos x\cdot\ln\left(\sin x\right)+3x\cdot\sin x\right]
(
sin
x
)
3
x
[
cos
x
⋅
ln
(
sin
x
)
+
3
x
⋅
sin
x
]
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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
.
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
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
)
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