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Practice: Chain Rule with Given Values
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Derivatives of Trig Functions
6 Activities
Wize University Calculus 1 Textbook > Derivatives
The Chain Rule
3 Activities
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
)
.
−
1
(
π
2
+
1
)
2
-\dfrac{1}{\bigg(\dfrac{\pi}{2}+1\bigg)^2}
−
(
2
π
+
1
)
2
1
1
2
\frac{1}{2}
2
1
1
2
−
1
2
\frac{1}{2}-\frac{1}{\sqrt{2}}
2
1
−
2
1
−
2
π
−
1
2
-\frac{2}{\pi}-\frac{1}{2}
−
π
2
−
2
1
I don't know
Check Submission
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
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
)
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.
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
))
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
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
.
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 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
))