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Higher Order Derivatives
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Higher Order Derivatives
4 Activities
Wize University Calculus 1 Textbook > Derivatives
Derivatives of Trig Functions
6 Activities
If
f
(
x
)
=
sin
(
2
x
)
f\left(x\right)=\sin\left(2x\right)
f
(
x
)
=
sin
(
2
x
)
, find
f
(
21
)
(
π
2
)
f^{\left(21\right)}\left(\frac{\pi}{2}\right)
f
(
21
)
(
2
π
)
.
(i.e. find the 21st derivative at the point
π
2
\frac{\pi}{2}
2
π
)
a.
2
20
2^{20}
2
20
b.
−
2
20
-2^{20}
−
2
20
c.
2
21
2^{21}
2
21
d.
−
2
21
-2^{21}
−
2
21
e. None of the above
I don't know
Check Submission
More Higher Order Derivatives Questions:
Higher Order Derivatives: The Power Rule
If
f
(
x
)
=
x
10
f\left(x\right)=x^{10}
f
(
x
)
=
x
10
find
f
(
9
)
(
x
)
f^{\left(9\right)}\left(x\right)
f
(
9
)
(
x
)
.
(i.e. the 9th derivative)
Higher Order Derivatives
Find the second derivative of
f
(
x
)
=
x
2
+
16
x
f(x) = x^2+16\sqrt{x}
f
(
x
)
=
x
2
+
16
x
.
Higher order derivatives
Consider the function
f
(
x
)
=
3
x
−
4
x
f(x)=\dfrac{3x-4}{x}
f
(
x
)
=
x
3
x
−
4
. Find the second derivative
f
′
′
(
x
)
,
at
x
=
1
f''(x),\text{ at }x=1
f
′′
(
x
)
,
at
x
=
1
.
Higher order derivatives
Consider the function
f
(
x
)
=
3
x
−
4
x
f(x)=\dfrac{3x-4}{x}
f
(
x
)
=
x
3
x
−
4
. Find the second derivative
f
′
′
(
x
)
,
at
x
=
1
f''(x),\text{ at }x=1
f
′′
(
x
)
,
at
x
=
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
Higher Order Derivatives
If
f
(
x
)
=
sin
(
2
x
)
f\left(x\right)=\sin\left(2x\right)
f
(
x
)
=
sin
(
2
x
)
, find
f
(
21
)
(
π
2
)
f^{\left(21\right)}\left(\frac{\pi}{2}\right)
f
(
21
)
(
2
π
)
.
(i.e. find the 21st derivative at the point
π
2
\frac{\pi}{2}
2
π
)
Higher Order Derivatives
If
f
(
x
)
=
sin
(
2
x
)
f\left(x\right)=\sin\left(2x\right)
f
(
x
)
=
sin
(
2
x
)
, find
f
(
21
)
(
π
2
)
f^{\left(21\right)}\left(\frac{\pi}{2}\right)
f
(
21
)
(
2
π
)
.
(i.e. find the 21st derivative at the point
π
2
\frac{\pi}{2}
2
π
)
Higher order derivatives
Consider the function
f
(
x
)
=
3
x
−
4
x
f(x)=\dfrac{3x-4}{x}
f
(
x
)
=
x
3
x
−
4
. Find the second derivative
f
′
′
(
x
)
,
at
x
=
1
f''(x),\text{ at }x=1
f
′′
(
x
)
,
at
x
=
1
.
Higher Order Derivatives
An object is thrown straight up in the air at
t
=
0
t = 0
t
=
0
seconds. Its height in metres at
t
t
t
seconds is given by
h
(
t
)
=
s
0
+
v
0
t
−
7
t
2
h(t) = s_0 + v_0 t - 7t^2
h
(
t
)
=
s
0
+
v
0
t
−
7
t
2
where
s
0
s_0
s
0
and
v
0
v_0
v
0
are constants. In the first second the object rises 7 metres. For how many seconds does the object rise before starting to fall back down?
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
Higher Order Derivatives: The Power Rule
If
f
(
x
)
=
x
10
f\left(x\right)=x^{10}
f
(
x
)
=
x
10
find
f
(
9
)
(
x
)
f^{\left(9\right)}\left(x\right)
f
(
9
)
(
x
)
.
(i.e. the 9th derivative)
Higher Order Derivatives
Find the third derivative of
f
(
x
)
=
x
2
+
16
x
f(x) = x^2+16\sqrt{x}
f
(
x
)
=
x
2
+
16
x
.
Higher order derivatives
Consider the function
f
(
x
)
=
3
x
−
4
x
f(x)=\dfrac{3x-4}{x}
f
(
x
)
=
x
3
x
−
4
. Find the second derivative
f
′
′
(
x
)
,
at
x
=
1
f''(x),\text{ at }x=1
f
′′
(
x
)
,
at
x
=
1
.
More Derivatives of Trig Functions Questions:
Quotient with Trig
Find the derivative of
h
(
t
)
=
cos
t
sin
t
(
1
+
cos
t
)
\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
h
(
t
)
=
sin
t
(
1
+
cos
t
)
cos
t
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Find the derivative of 𝑓(𝑥) = 𝑒
𝑥
tan 𝑥 + 3.
Differentiation Rules
Find
d
d
x
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
d
x
d
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
Differentiation Rules
Find
d
d
x
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
d
x
d
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Derivatives: Trigonometric Functions
Find
f
′
(
π
)
+
f
′
′
(
π
)
f'(\pi)+f''(\pi)
f
′
(
π
)
+
f
′′
(
π
)
if
𝑓
(
𝑥
)
=
𝑥
cos
𝑥
−
sin
𝑥
𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥
f
(
x
)
=
x
cos
x
−
sin
x
.
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Differentiation Rules
Find
d
d
x
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
d
x
d
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
Differentiation Rules
Find
d
d
x
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
d
x
d
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
Higher Order Trig Derivative
Find
y
23
y^{23}
y
23
given
y
=
sin
x
\displaystyle y=\sin x
y
=
sin
x
Higher Order Derivatives
If
f
(
x
)
=
sin
(
2
x
)
f\left(x\right)=\sin\left(2x\right)
f
(
x
)
=
sin
(
2
x
)
, find
f
(
21
)
(
π
2
)
f^{\left(21\right)}\left(\frac{\pi}{2}\right)
f
(
21
)
(
2
π
)
.
(i.e. find the 21st derivative at the point
π
2
\frac{\pi}{2}
2
π
)
Differentiating functions
Find the derivative of the function
f
(
x
)
=
sin
(
x
)
cos
(
x
)
x
3
f(x) = \frac{\sin(x)\cos(x)}{x^3}
f
(
x
)
=
x
3
sin
(
x
)
cos
(
x
)
You do not need to simplify your answer.
Differentiation Rules
Find
d
d
x
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
d
x
d
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of
cos
4
x
+
x
4
4
x
\cos^4x+\frac{x^4}{4^x}
cos
4
x
+
4
x
x
4
.
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of
cos
4
x
+
x
4
4
x
\cos^4x+\frac{x^4}{4^x}
cos
4
x
+
4
x
x
4
.
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
)
Higher Order Derivatives
If
f
(
x
)
=
sin
(
2
x
)
f\left(x\right)=\sin\left(2x\right)
f
(
x
)
=
sin
(
2
x
)
, find
f
(
21
)
(
π
2
)
f^{\left(21\right)}\left(\frac{\pi}{2}\right)
f
(
21
)
(
2
π
)
.
(i.e. find the 21st derivative at the point
π
2
\frac{\pi}{2}
2
π
)
Derivatives: Trigonometric Functions
Find the equation of the tangent line to
f
(
x
)
=
(
x
2
−
2
)
sin
x
+
2
x
cos
x
f(x) = (x^2 - 2) \sin x+2x \cos x
f
(
x
)
=
(
x
2
−
2
)
sin
x
+
2
x
cos
x
at
x
=
π
x = \pi
x
=
π
.
Derivatives: Trigonometric Functions
Find
f
′
(
π
)
+
f
′
′
(
π
)
f'(\pi)+f''(\pi)
f
′
(
π
)
+
f
′′
(
π
)
if
𝑓
(
𝑥
)
=
𝑥
cos
𝑥
−
sin
𝑥
𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥
f
(
x
)
=
x
cos
x
−
sin
x
.
Derivatives: Trigonometric Functions
Find the derivative of the following function:
f
(
x
)
=
sin
x
cos
x
+
x
3
f(x)=\sin x\cos x+\sqrt[3]{x}
f
(
x
)
=
sin
x
cos
x
+
3
x
Derivatives: Trigonometric Functions
Given that
𝑓
(
𝑥
)
=
𝑥
sin
𝑥
𝑓(𝑥) = 𝑥\sin 𝑥
f
(
x
)
=
x
sin
x
, find
f
′
(
3
π
2
)
f'\ \left(\dfrac{3\pi}{2}\right)
f
′
(
2
3
π
)
.
Derivatives: Trigonometric Functions
Find the derivative of
g
(
x
)
=
sin
(
π
x
+
1
)
−
cos
(
π
x
+
3
2
)
g\left(x\right)=\sin\left(\pi x+1\right)-\cos\left(\dfrac{\pi x+3}{2}\right)
g
(
x
)
=
sin
(
π
x
+
1
)
−
cos
(
2
π
x
+
3
)
Derivatives: Trigonometric Functions
Find the derivative of
f
(
x
)
=
3
x
2
cos
(
2
x
)
\ f\left(x\right)=3x^2\cos\left(2x\right)
f
(
x
)
=
3
x
2
cos
(
2
x
)
.
Derivatives: Exponential and Trigonometric Functions
Differentiate
y
=
csc
(
1
−
3
5
x
)
y=\csc(1-3^{5\sqrt x})
y
=
csc
(
1
−
3
5
x
)
Product and Chain
Q:
\textbf{Q:}
Q:
Find the derivative of
h
(
θ
)
=
θ
2
−
1
cos
(
3
θ
−
1
)
e
−
5
θ
h(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
h
(
θ
)
=
θ
2
−
1
cos
(
3
θ
−
1
)
e
−
5
θ
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
)
.
Second Derivative with Trig
Find the second derivative for the function
g
(
s
)
=
1
tan
s
+
2
e
s
−
cot
s
cos
s
\displaystyle g(s)=\frac{1}{\tan s} +2e^{s}-\frac{\cot s}{\cos s}
g
(
s
)
=
tan
s
1
+
2
e
s
−
cos
s
cot
s
Higher Order Trig Derivative
Find
y
23
y^{23}
y
23
given
y
=
sin
x
\displaystyle y=\sin x
y
=
sin
x
Quotient with Trig
Find the derivative of
h
(
t
)
=
cos
t
sin
t
(
1
+
cos
t
)
\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
h
(
t
)
=
sin
t
(
1
+
cos
t
)
cos
t
Tangent Line with Product Rule
Find the equation of the tangent line to the graph of
𝑦
=
sin
𝑥
+
3
𝑥
2
cos
𝑥
𝑦 = \sin 𝑥 + 3𝑥^2 \cos 𝑥
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
\displaystyle x=\frac{\pi}{2}
x
=
2
π
Product and Chain
Q:
\textbf{Q:}
Q:
Find the derivative of
h
(
θ
)
=
θ
2
−
1
cos
(
3
θ
−
1
)
e
−
5
θ
h(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
h
(
θ
)
=
θ
2
−
1
cos
(
3
θ
−
1
)
e
−
5
θ
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f
(
x
)
=
e
x
2
sec
(
x
)
\displaystyle f(x)=e^{x^2\text{sec}(x)}
f
(
x
)
=
e
x
2
sec
(
x
)
Find the derivative of the following function.
f
(
x
)
=
x
2
+
1
+
cos
x
\displaystyle f(x)=\sqrt{x^2+1+\cos{x}}
f
(
x
)
=
x
2
+
1
+
cos
x
Derivatives: Trigonometric Functions
Find the slope of the tangent line to the following function at
x
=
π
/
6
:
x=\pi/6:
x
=
π
/6
:
g
(
x
)
=
e
−
1
2
ln
(
sin
x
)
g(x)=e^{-\frac{1}{2}\ln(\sin x)}
g
(
x
)
=
e
−
2
1
l
n
(
s
i
n
x
)
Derivatives: Trigonometric Functions
Given that
g
(
1
)
=
g
′
(
1
)
=
1
g(1)=g'(1)=1
g
(
1
)
=
g
′
(
1
)
=
1
, find
f
′
(
π
/
2
)
if
f
(
x
)
=
g
(
sin
x
)
x
+
1
f'(\pi/2) \text{ if }f(x)=\frac{\sqrt{g(\sin x)}}{x+1}
f
′
(
π
/2
)
if
f
(
x
)
=
x
+
1
g
(
s
i
n
x
)
Derivatives: Trigonometric Functions
Find
d
22
d
x
22
(
−
10
cos
(
x
)
)
\frac{d^{22}}{dx^{22}}(-10\cos(x))
d
x
22
d
22
(
−
10
cos
(
x
))
Derivatives: Trigonometric Functions
Find the equation of the tangent line and the normal line to
f
(
x
)
=
(
x
2
−
2
)
sin
x
+
2
x
cos
x
f(x)=(x^2-2)\sin x+2x\cos x
f
(
x
)
=
(
x
2
−
2
)
sin
x
+
2
x
cos
x
at
x
=
π
x=\pi
x
=
π
Derivatives: Trigonometric and Exponential Functions
Compute the derivative of
f
(
x
)
=
tan
(
e
2
x
)
f(x)=\tan(e^{2x})
f
(
x
)
=
tan
(
e
2
x
)
Derivatives: Trigonometric Functions
Find the equation of the line tangent to the graph of
f
(
x
)
=
x
3
sin
(
π
x
)
f(x)=x^3\sin(\pi x)
f
(
x
)
=
x
3
sin
(
π
x
)
at the point
x
=
2.
x=2.
x
=
2.
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
)
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Find the equation of the tangent line of the graph
y
=
e
x
cos
x
y=\frac{e^x}{\cos\ x}
y
=
c
o
s
x
e
x
at the point where 𝑥 = 0.
Derivatives: Trigonometric Functions
Find
f
′
(
π
)
+
f
′
′
(
π
)
f'(\pi)+f''(\pi)
f
′
(
π
)
+
f
′′
(
π
)
if
𝑓
(
𝑥
)
=
𝑥
cos
𝑥
−
sin
𝑥
𝑓(𝑥) = 𝑥 \cos 𝑥 − \sin 𝑥
f
(
x
)
=
x
cos
x
−
sin
x
.
Derivatives: Trigonometric Functions
Find the derivative of the following function:
f
(
x
)
=
sin
x
cos
x
+
x
3
f(x)=\sin x\cos x+\sqrt[3]{x}
f
(
x
)
=
sin
x
cos
x
+
3
x
Derivatives: Trigonometric Functions
Given that
𝑓
(
𝑥
)
=
𝑥
sin
𝑥
𝑓(𝑥) = 𝑥\sin 𝑥
f
(
x
)
=
x
sin
x
, find
f
′
(
3
π
2
)
f'\ \left(\dfrac{3\pi}{2}\right)
f
′
(
2
3
π
)
.
Derivatives: Trigonometric Functions
Find the derivative of
g
(
x
)
=
sin
(
π
x
+
1
)
−
cos
(
π
x
+
3
2
)
g\left(x\right)=\sin\left(\pi x+1\right)-\cos\left(\dfrac{\pi x+3}{2}\right)
g
(
x
)
=
sin
(
π
x
+
1
)
−
cos
(
2
π
x
+
3
)
Derivatives: Trigonometric Functions
Find the equation of the tangent line to
f
(
x
)
=
(
x
2
−
2
)
sin
x
+
2
x
cos
x
f(x) = (x^2 - 2) \sin x+2x \cos x
f
(
x
)
=
(
x
2
−
2
)
sin
x
+
2
x
cos
x
at
x
=
π
x = \pi
x
=
π
.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Practice: Equation of Tangent Line
Practice: Equation of Tangent Line
Find the equation of the tangent line to the graph
y
=
sin
x
+
3
x
2
cos
x
y=\sin x+3x^2\cos x
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
x=\frac{\pi}{2}
x
=
2
π
.
Derivatives: Trigonometric Functions
Find the derivative of
f
(
x
)
=
3
x
2
cos
(
2
x
)
\ f\left(x\right)=3x^2\cos\left(2x\right)
f
(
x
)
=
3
x
2
cos
(
2
x
)
.
Derivatives: Exponential and Trigonometric Functions
Differentiate
y
=
csc
(
1
−
3
5
x
)
y=\csc(1-3^{5\sqrt x})
y
=
csc
(
1
−
3
5
x
)
Product and Chain
Q:
\textbf{Q:}
Q:
Find the derivative of
h
(
θ
)
=
θ
2
−
1
cos
(
3
θ
−
1
)
e
−
5
θ
h(\theta)=\sqrt{\theta^2-1}\cos(3\theta-1)e^{-5\theta}
h
(
θ
)
=
θ
2
−
1
cos
(
3
θ
−
1
)
e
−
5
θ
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
)
.
Higher Order Trig Derivative
Find
y
23
y^{23}
y
23
given
y
=
sin
x
\displaystyle y=\sin x
y
=
sin
x
Second Derivative with Trig
Find the second derivative for the function
g
(
s
)
=
1
tan
s
+
2
e
s
−
cot
s
cos
s
\displaystyle g(s)=\frac{1}{\tan s} +2e^{s}-\frac{\cot s}{\cos s}
g
(
s
)
=
tan
s
1
+
2
e
s
−
cos
s
cot
s
Quotient with Trig
Find the derivative of
h
(
t
)
=
cos
t
sin
t
(
1
+
cos
t
)
\displaystyle h(t)=\frac{\cos t}{\sin t(1+\cos t)}
h
(
t
)
=
sin
t
(
1
+
cos
t
)
cos
t
Tangent Line with Product Rule
Find the equation of the tangent line to the graph of
𝑦
=
sin
𝑥
+
3
𝑥
2
cos
𝑥
𝑦 = \sin 𝑥 + 3𝑥^2 \cos 𝑥
y
=
sin
x
+
3
x
2
cos
x
at the point where
x
=
π
2
\displaystyle x=\frac{\pi}{2}
x
=
2
π
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
)
Derivatives of Trigonometric Functions
Determine the point(s), if any, where the function
y
=
cos
2
x
−
sin
2
x
y=\cos^2x-\sin^2x
y
=
cos
2
x
−
sin
2
x
has a horizontal tangent line.
Derivatives: Trigonometric and Logarithmic Functions
Find
d
d
y
[
(
y
5
+
ln
y
)
tan
y
]
\displaystyle \frac{\text{d}}{\text{d}y}\left[\left(y^5+\ln y\right)\tan y\right]
d
y
d
[
(
y
5
+
ln
y
)
tan
y
]
.
Derivatives: Trigonometric and Exponential Functions
Evaluate
d
d
x
(
e
x
sin
x
)
\displaystyle \frac{\text{d}}{\text{d}x}\left(e^x\sin x\right)
d
x
d
(
e
x
sin
x
)
.
Find the derivative of the function
f
(
x
)
=
sin
x
cos
x
+
x
3
f(x)=\sin x\cos x+\sqrt[3]{x}
f
(
x
)
=
sin
x
cos
x
+
3
x
.
Find the equation of the tangent line to
f
(
x
)
=
(
x
2
−
2
)
sin
x
+
2
x
cos
x
f(x) = (x^2 - 2) \sin x+2x \cos x
f
(
x
)
=
(
x
2
−
2
)
sin
x
+
2
x
cos
x
at
x
=
π
x = \pi
x
=
π
.
Find the derivative of 𝑓(𝑥) = 𝑒
𝑥
tan 𝑥 + 3.
Given the function
f
(
x
)
=
g
(
x
)
h
(
x
)
−
e
x
cos
x
f\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}-e^x\cos x
f
(
x
)
=
h
(
x
)
g
(
x
)
−
e
x
cos
x
,
g
(
π
2
)
=
2
g\left(\frac{\pi}{2}\right)=2
g
(
2
π
)
=
2
,
g
′
(
π
2
)
=
3
g'\left(\frac{\pi}{2}\right)=3
g
′
(
2
π
)
=
3
,
h
(
π
2
)
=
1
h\left(\frac{\pi}{2}\right)=1
h
(
2
π
)
=
1
, and
h
′
(
π
2
)
=
2
h'\left(\frac{\pi}{2}\right)=2
h
′
(
2
π
)
=
2
, find
f
′
(
π
2
)
f'\left(\frac{\pi}{2}\right)
f
′
(
2
π
)
.
Finding a derivative: Trigonometric and Exponential Functions
Find the derivative of
cos
4
x
+
x
4
4
x
\cos^4x+\frac{x^4}{4^x}
cos
4
x
+
4
x
x
4
.
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
)
Differentiating functions
Find the derivative of the function
f
(
x
)
=
sin
(
x
)
cos
(
x
)
x
3
f(x) = \frac{\sin(x)\cos(x)}{x^3}
f
(
x
)
=
x
3
sin
(
x
)
cos
(
x
)
You do not need to simplify your answer.
Practice: Chain Rule
Find the derivative of the following function:
y
=
sin
(
x
)
y=\sin(\sqrt{x})
y
=
sin
(
x
)
Differentiation Rules
Find
d
d
x
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
\frac{d}{dx}(\sin^2{x}+\sqrt\pi+\cos^2{x}+5^7)
d
x
d
(
sin
2
x
+
π
+
cos
2
x
+
5
7
)
Derivatives: Trigonometric Functions
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
)
, where
g
(
1
)
=
g
′
(
1
)
=
1.
g(1)=g^{\prime}(1)=1.
g
(
1
)
=
g
′
(
1
)
=
1.
Derivatives: Trigonometric Functions
Find the derivative of the following function.
f
(
x
)
=
x
2
+
1
+
cos
x
\displaystyle f(x)=\sqrt{x^2+1+\cos{x}}
f
(
x
)
=
x
2
+
1
+
cos
x
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
))