This page breaks down the derivatives of inverse trigonometric functions such as arcsin, arccos, arctan, arccot, arccsc, and arcsec. You’ll find a formula reference sheet, and many practice problems with answers to help you master this essential calculus skill.

 

Notes

Practice Problems

Find the derivative of each

\(\textbf{1)}\) \(f(x)=\cos^2(x)+3\sin^{−1}(x), \text{find } f'(x)\)

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The derivative is \(f'(x)=-2\sin(x)\cos(x)+\displaystyle \frac{3}{\sqrt{1-x^2}}\)

 

\(\textbf{2)}\) \(f(x)=8\sin^{−1}(x)−2\csc^{−1}(x), \text{find } f'(x)\)

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The derivative is \(f'(x)=\displaystyle \frac{8}{\sqrt{1-x^2}}+\frac{2}{|x|\sqrt{x^2-1}}\)

 

\(\textbf{3)}\) \(f(x)=\arctan(2x)+4\tan(3x), \text{find } f'(x)\)

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The derivative is \(f'(x)=\displaystyle\frac{2}{4x^2+1}+12\sec ^2\left(3x\right)
\)

 

\(\textbf{4)}\) \(f(x)=\sec^{−1}(x)−\cos^{−1}(x), \text{find } f'(x)\)

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The derivative is \(f'(x) =\displaystyle\frac{1}{|x|\sqrt{x^2-1}}+\frac{1}{\sqrt{1-x^2}}\)

 

\(\textbf{5)}\) \(f(x)=x^3\arcsin(2x), \text{find } f'(x)\)

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The derivative is \(f'(x) =3x^2\arcsin \left(2x\right)+\displaystyle\frac{2x^3}{\sqrt{1-4x^2}}\)

 

\(\textbf{6)}\) \(f(x)=\displaystyle \frac{\sin^{−1}(x)}{x}, \text{find } f'(x)\)

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The derivative is \(f'(x) =\displaystyle\frac{\frac{x}{\sqrt{1-x^2}}-\sin^{-1} \left(x\right)}{x^2}=\frac{x-\sin^{-1} \left(x\right)\sqrt{1-x^2}}{x^2\sqrt{1-x^2}}\)

 

\(\textbf{7)}\) \(f(x)=3x^5+\tan^{−1}(3x^5), \text{find } f'(x)\)

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The derivative is \(f'(x) =15x^4+\displaystyle\frac{15x^4}{9x^{10}+1}\,\,= \,\,\displaystyle \frac{135x^{14}+30x^4}{9x^{10}+1}\)

 

\(\textbf{8)}\) \(f(x)=\sin^{−1}(\ln x), \text{find } f'(x)\)

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The derivative is \(f'(x) =\displaystyle \frac{1}{x\sqrt{1-\left(\ln x \right)^2}}\)

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\(\,\,\,\,\,f(x)=\sin^{−1}(\ln x)\)
\(\,\,\,\,\,u=\ln x, u’=\frac{1}{x}\)
\(\,\,\,\,\,\frac{d}{dx} \arcsin{u}=\displaystyle \frac{u’}{\sqrt{1-u^2}}\)
\(\,\,\,\,\,\frac{d}{dx} \arcsin{u}=\displaystyle \frac{\frac{1}{x}}{\sqrt{1-(\ln{x})^2}}\)
\(\,\,\,\,\,f'(x) =\displaystyle \frac{1}{x\sqrt{1-\left(\ln x \right)^2}}\)

 

\(\textbf{9)}\) \(f(x)=\ln (\arcsin(x)), \text{find } f'(x)\)

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The derivative is \(f'(x) =\displaystyle \frac{1}{\arcsin \left(x\right)\sqrt{1-x^2}}\)

 

\(\textbf{10)}\) \(f(x) = \cot^{-1}(2x), \text{find } f'(x)\)

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The derivative is \(f'(x) =-\displaystyle \frac{2}{4x^2+1}\)

 

\(\textbf{11)}\) \(y = \tan^{-1}(x^2+4x), \text{find } \displaystyle\frac{dy}{dx}\)

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The derivative is \(\displaystyle\frac{dy}{dx} =\displaystyle \frac{2x+4}{1+\left(x^2+4x\right)^2}\)

 

 

See Related Pages\(\)

\(\bullet\text{ Inverse Trig Derivative Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)

\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)

\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)

\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)

\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)

\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)

\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)

\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)

\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)

\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)

\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)

\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

In Summary

Inverse trigonometric functions are first introduced to solve problems involving unknown angles but known sides in right triangles. These functions include the inverse sine (arcsin), inverse cosine (arccos), and inverse tangent (arctan). Most people tend to memorize the inverse trig derivatives.

\(\displaystyle \frac{d}{dx} \arcsin(x)=\displaystyle \frac{1}{\sqrt{1-x^2}} \,\,\,\,\, x \ne \pm1 \) \(\displaystyle \frac{d}{dx} \arccos(x)=\frac{-1}{\sqrt{1-x^2}} \,\,\,\,\, x \ne \pm1 \) \(\displaystyle \frac{d}{dx} \arctan(x)=\frac{1}{1+x^2} \) \(\displaystyle \frac{d}{dx} \text{arccot}(x)=\frac{-1}{1+x^2} \) \(\displaystyle \frac{d}{dx} \text{arcsec}(x)= \frac{1}{|x|\sqrt{x^2-1}} \,\,\,\,\, x \ne \pm1 \) \(\displaystyle \frac{d}{dx} \text{arccsc}(x)=\frac{-1}{|x|\sqrt{x^2-1}} \,\,\,\,\, x \ne \pm1 \)