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Derivatives: Inverse Trigonometric Functions
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Derivatives of Inverse Trig Functions
3 Activities
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
)
−
(
arcsin
(
x
)
)
−
2
⋅
1
1
−
x
2
+
1
1
−
(
1
x
)
2
⋅
1
x
2
-(\arcsin(x))^{-2}\cdot\frac{1}{\sqrt{1-x^2}}+\frac{1}{\sqrt{1-\left(\frac{1}{x}\right)^2}}\cdot\frac{1}{x^2}
−
(
arcsin
(
x
)
)
−
2
⋅
1
−
x
2
1
+
1
−
(
x
1
)
2
1
⋅
x
2
1
−
(
arcsin
(
x
)
)
−
2
⋅
1
1
−
x
2
+
1
1
−
(
1
x
)
2
⋅
−
1
x
2
-(\arcsin(x))^{-2}\cdot\dfrac{1}{\sqrt{1-x^2}}+\dfrac{1}{\sqrt{1-\left(\frac{1}{x}\right)^2}}\cdot\dfrac{-1}{x^2}
−
(
arcsin
(
x
)
)
−
2
⋅
1
−
x
2
1
+
1
−
(
x
1
)
2
1
⋅
x
2
−
1
(
arcsin
(
x
)
)
−
2
⋅
1
1
−
x
2
+
1
1
−
(
1
x
)
2
⋅
−
1
x
2
(\arcsin(x))^{-2}\cdot\dfrac{1}{\sqrt{1-x^2}}+\dfrac{1}{\sqrt{1-\left(\frac{1}{x}\right)^2}}\cdot\dfrac{-1}{x^2}
(
arcsin
(
x
)
)
−
2
⋅
1
−
x
2
1
+
1
−
(
x
1
)
2
1
⋅
x
2
−
1
(
arcsin
(
x
)
)
−
2
⋅
1
1
−
x
2
+
1
1
−
(
1
x
)
2
⋅
1
x
2
(\arcsin(x))^{-2}\cdot\frac{1}{\sqrt{1-x^2}}+\frac{1}{\sqrt{1-\left(\frac{1}{x}\right)^2}}\cdot\frac{1}{x^2}
(
arcsin
(
x
)
)
−
2
⋅
1
−
x
2
1
+
1
−
(
x
1
)
2
1
⋅
x
2
1
I don't know
Check Submission
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
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
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.
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
))