Tutorial 7: Derivative of Inverse Trigonometric Functions

Derivatives of Inverse Trigonometric Functions

Standard Derivatives

Function	Derivative
$\frac{d}{dx}[\sin^{-1} x]$	$\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}[\cos^{-1} x]$	$-\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}[\tan^{-1} x]$	$\frac{1}{1+x^2}$
$\frac{d}{dx}[\csc^{-1} x]$	$-\frac{1}{
$\frac{d}{dx}[\sec^{-1} x]$	$\frac{1}{
$\frac{d}{dx}[\cot^{-1} x]$	$-\frac{1}{1+x^2}$

Chain Rule Application

• $\frac{d}{dx}[\sin^{-1}(u)] = \frac{u'}{\sqrt{1-u^2}}$
• $\frac{d}{dx}[\cos^{-1}(u)] = -\frac{u'}{\sqrt{1-u^2}}$
• $\frac{d}{dx}[\tan^{-1}(u)] = \frac{u'}{1+u^2}$
• $\frac{d}{dx}[\sec^{-1}(u)] = \frac{u'}{|u|\sqrt{u^2-1}}$

Examples

First Derivatives

• $y = \sin^{-1}(3x)$: $y' = \frac{3}{\sqrt{1-9x^2}}$
• $y = \cos^{-1}\left(\frac{x+1}{2}\right)$: $y' = -\frac{1}{2\sqrt{1-\left(\frac{x+1}{2}\right)^2}}$
• $y = \sin^{-1}\left(\frac{1}{x}\right)$: $y' = -\frac{1}{|x|\sqrt{x^2-1}}$
• $y = \sec^{-1}(x^2)$: $y' = \frac{2}{|x|\sqrt{x^4-1}}$
• $y = \tan^{-1}(e^{2x})$: $y' = \frac{2e^{2x}}{1+e^{4x}}$

Combined Functions

• $y = x + \sin^{-1}(e^{-x})$: $y' = 1 - \frac{e^{-x}}{\sqrt{1-e^{-2x}}}$
• $y = \tan^{-1}(x^2) \csc^{-1}(\ln x)$: Use product rule
• $y = \frac{\cos^{-1}(2x)}{3x-e^{2x}}$: Use quotient rule

Hyperbolic Functions (Review)

• Identities: $\cosh x + \sinh x = e^x$
• Definitions: $\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$, $\text{sech } x = \frac{2}{e^x + e^{-x}}$
• Identity: $\text{sech}^2 x = 1 - \tanh^2 x$