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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Inverse Trig Functions

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Derivatives of Inverse Trig Functions

The derivatives of the six inverse trigonometric functions are:

(arcsin⁡(x))′=11−x2       (arccos⁡(x))′=−11−x2 (arctan⁡(x))′=11+x2  (arcsec(x))′=1∣x∣x2−1 (arcCot(x))′=−11+x2(arcCsc(x))′=−1∣x∣x2−1\boxed{\quad (\arcsin(x))' =\frac{1}{\sqrt{1-x^2}}\qquad \ \ \ \ \ \ \hspace{0.25in}\ (\arccos(x))'= \frac{-1}{\sqrt{1-x^2}}\quad }\\ \\\text{ }\\ \boxed{\quad (\arctan(x))' =\frac{1}{{1+x^2}} \ \ \qquad\hspace{0.16in}(\arcsec (x))'=\frac{1}{|x|\sqrt{x^2-1}}\quad }\\ \\\text{ }\\ \boxed{\quad (\arccot(x))' =\frac{-1}{{1+x^2}}\qquad\hspace{0.03in}(\arccsc (x))'=\frac{-1}{|x|\sqrt{x^2-1}}\quad }










Wize Tip
Notice how all the derivatives of "CO Functions" have a minus sign.

Wize Concept
The Product, Quotient, and Chain Rules still apply for Inverse Trig Derivatives!

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Example: Inverse Trig Derivative

Find the derivative of f(x)=arcsin⁡(3x2)f(x)=\arcsin(3x^2).

f′(x)=[11−(3x2)2]×[3(2x)]\displaystyle f'\left(x\right)=\left[\frac{1}{\sqrt{1-\left(3x^2\right)^2}}\right]\times\left[3\left(2x\right)\right]

f′(x)=6x1−9x4f'(x)=\displaystyle \frac{6x}{\sqrt{1-9x^4}}
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Find the derivative of f(x)=(arccos⁡x)2sec⁡x\displaystyle f\left(x\right)=\frac{\left(\arccos x\right)^2}{\sec x}


Extra Practice
Inverse trigonometric derivative
Find the derivative of g(x)=(cos⁡−1x)(sin⁡−1x)g(x)=\left(\cos^{-1}x\right)\left(\sin^{-1}x\right).
Derivatives: Inverse Trigonometric Functions
Given that f(x)=2−arcsin⁡(x)f\left(x\right)=2-\arcsin\left(x\right), find f′(0)f'\left(0\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin⁡(x+2)f\left(x\right)=\arcsin\left(x+2\right)
Derivatives: Inverse Trigonometric Functions
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right)
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos⁡−1(x)−2025)′(3^{3x-x^{4}}+2^{\cos^{-1}(x)}-2025)'
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Finding a Derivative
If f(x)=tan⁡−1(3sin⁡x)f(x)=\tan^{-1}\left(3^{\sin x}\right), find f′(π)f'(\pi).
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Inverse Trigonometric Functions
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{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⁡(3x2)f(x)=\arcsin(3x^2)
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Derivatives: Inverse Trigonometric Functions
Find [tan⁡−1(x3)]′[\tan^{-1}(\sqrt[3]x)]'
Derivatives: Inverse Trigonometric Functions
Show thatddxsin⁡−1x=11−x2\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
Practice: Derivative of Inverse Trig
Q:\textbf{Q:} Show that ddxcos⁡−1x=−11−x2.\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
Practice: Tangent Line with Inverse Trig
Q:\textbf{Q:} Find the equation of the tangent line to the graph of f(x)=cot⁡−1(x2)\displaystyle f(x)=\cot^{-1}(x^{2}) at the point x=2x=2.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Derivatives: Exponential and Trigonometric Functions
Find the derivative of
f(x)=5x6−sec⁡−1(e2x)+cos⁡−1xf(x)=5x^6-\sec^{-1}(e^{2x})+\cos^{-1}x
Inverse Trig Derivative
Find the equation of the line tangent to
f(x)=sin⁡−1x1+xf(x)=\frac{\sin^{-1}x}{1+x}
at x=1/2x=1/2.
Derivatives: Inverse Trigonometric Functions
Find f′(x)f'(x) if f(x)=arctan⁡xf(x)=\arctan \sqrt{x}
Differentiate the following
y=tan⁡−1x2−1y=\tan^{-1}\sqrt{x^2-1}
Find the derivative of the following function.
f(x)=arcsinx\displaystyle f(x)=\sqrt{\text{arcsin{x}}}
The Chain Rule
Q:\bf{Q:} Find (33x−x4+2cos−1(x)−2019)′(3^{3x-x^4}+2^{cos^{-1}(x)}-2019)'
Derivatives: Inverse Trigonometric Functions
Find [tan⁡−1(x3)]′[\tan^{-1}(\sqrt[3]x)]'
Derivatives: Inverse Trigonometric Functions
Show thatddxsin⁡−1x=11−x2\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}
Practice: Derivative of Inverse Trig
Q:\textbf{Q:} Show that ddxcos⁡−1x=−11−x2.\displaystyle\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}.
Practice: Tangent Line with Inverse Trig
Q:\textbf{Q:} Find the equation of the tangent line to the graph of f(x)=cot⁡−1(x2)\displaystyle f(x)=\cot^{-1}(x^{2}) at the point x=2x=2.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Inverse Trigonometric Functions
Compute the derivative of f(x)=arcsin(x+2x−1)\displaystyle f(x)=\text{arcsin}\left(\frac{x+2}{x-1}\right)
Derivatives: Exponential Functions, Inverse Trigonometric Functions
Compute the derivative of f(x)=sec⁡(32x)f(x)=\sec(3^{2x}) .
Derivatives: Logarithmic and Inverse Trigonometric Functions
Evaluate ddx(sin⁡−1xsin⁡xlog⁡3x)\displaystyle \frac{\text{d}}{\text{d}x}\left( \frac{\sin^{-1}x\sin x}{\log_3x}\right).
Find the derivative of f(x)=arcsin⁡(ex+1)f(x)=\arcsin(e^{x+1}).
Inverse trigonometric derivative
Find the derivative of g(x)=cos⁡−1(sin⁡−1x)g(x)=\cos^{-1}\left(\sin^{-1}x\right).
Derivatives: Inverse Trigonometric Functions
Find the derivative of f(x)=(arccos⁡x)2sec⁡xf\left(x\right)=\frac{\left(\arccos x\right)^2}{\sec x}.
Evaluating a derivative at a point
Evaluate (cos⁡−1(x2))′∣0\left( \cos^{-1}\left(x^2\right) \right)'\rvert_{0}
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Find ddx(arctan⁡(etan⁡x))\frac{d}{dx}\left(\arctan\left(e^{\tan x}\right)\right).
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⁡(3x2)f(x)=\arcsin(3x^2)
Derivatives: Inverse Trigonometric Functions
Practice: Finding a Derivative
Given that f(x)=(2−arcsin⁡(x2))3f\left(x\right)=\left(2-\arcsin\left(x^2\right)\right)^3, find f′(0)f'\left(0\right).
Finding a Derivative
Practice: Finding a Derivative
(tan⁡−1(3sin⁡x))′∣x=π=\left(\tan^{-1}\left(3^{\sin x}\right)\right)'|_{x=\pi}=
Linear Approximation and Trigonometric Derivative
a) Show that for f(x)=arcsin⁡(x)f(x) = \arcsin(x) , that f′(x)=11−x2f'(x) = \frac{1}{\sqrt{1 - x^2}} .
b) Use linear approximation at a suitable close value to estimate 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)=1arcsin(x)+arcsin(1x)\displaystyle f(x)=\frac{1}{\text{arcsin}(x)}+\text{arcsin}\left(\frac{1}{x}\right)
Inverse Trigonometric Functions
Calculate the derivative of the following functions.
f(x)=arcsin(ex+1)\displaystyle f(x)=\text{arcsin}(e^{x+1})
Derivatives: Trigonometric and Exponential Functions
Calculate the derivative of the following functions.
f(x)=tan⁡(arccos(e4x))\displaystyle f(x)=\tan(\text{arccos}(e^{4x}))