Differentiating inverse trigonometric functions

The derivatives of inverse sine, inverse cosine and inverse tangent and their proofs, the standard forms with x/a, using the chain and product rules, and tangents to inverse trigonometric graphs.

The derivatives

Standard derivatives

$$\frac{d}{dx}\sin^{-1}\frac{x}{a} = \frac{1}{\sqrt{a^2 - x^2}} \qquad \frac{d}{dx}\cos^{-1}\frac{x}{a} = \frac{-1}{\sqrt{a^2 - x^2}} \qquad \frac{d}{dx}\tan^{-1}\frac{x}{a} = \frac{a}{a^2 + x^2}$$

With $a = 1$: the derivatives of $\sin^{-1}x$, $\cos^{-1}x$ and $\tan^{-1}x$ are $\dfrac{1}{\sqrt{1 - x^2}}$, $\dfrac{-1}{\sqrt{1 - x^2}}$ and $\dfrac{1}{1 + x^2}$.

Example 1: proof for inverse sine●●●

Prove that $\dfrac{d}{dx}\sin^{-1}x = \dfrac{1}{\sqrt{1 - x^2}}$.

Let $y = \sin^{-1}x$, so $x = \sin y$ with $-\frac{\pi}{2} \le y \le \frac{\pi}{2}$.

$\dfrac{dx}{dy} = \cos y$. In this range $\cos y \ge 0$, so $\cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - x^2}$.

So $\dfrac{dy}{dx} = \dfrac{1}{\sqrt{1 - x^2}}$.

With the chain and product rules

Example 2●●

Differentiate $\tan^{-1}3x$, $\sin^{-1}(x^2)$ and $x\sin^{-1}x$.

$\dfrac{d}{dx}\tan^{-1}3x = \dfrac{3}{1 + 9x^2}$.

$\dfrac{d}{dx}\sin^{-1}(x^2) = \dfrac{2x}{\sqrt{1 - x^4}}$.

$\dfrac{d}{dx}\big(x\sin^{-1}x\big) = \sin^{-1}x + \dfrac{x}{\sqrt{1 - x^2}}$.

Example 3: a tangent●●

Find the tangent to $y = \tan^{-1}x$ at $x = 1$.

The point is $\big(1, \frac{\pi}{4}\big)$ and the gradient is $\dfrac{1}{1 + 1} = \dfrac12$.

$y - \frac{\pi}{4} = \frac12(x - 1)$, so $y = \frac12 x - \frac12 + \frac{\pi}{4}$.

Flashcards

$dfrac{d}{dx}sin^{-1}x$

$\dfrac{1}{\sqrt{1 - x^2}}$

$dfrac{d}{dx}cos^{-1}x$

$\dfrac{-1}{\sqrt{1 - x^2}}$

$dfrac{d}{dx} an^{-1}x$

$\dfrac{1}{1 + x^2}$

$dfrac{d}{dx} an^{-1}dfrac{x}{a}$

$\dfrac{a}{a^2 + x^2}$

Practice

Find the derivative of $\tan^{-1}3x$ at $x = 0$.
Find the derivative of $\sin^{-1}\dfrac{x}{5}$ at $x = 3$.
Find the gradient of $y = \tan^{-1}x$ at $x = 1$.
Find the derivative of $\cos^{-1}\dfrac{x}{2}$ at $x = 0$.
Find the derivative of $\sin^{-1}2x$ at $x = 0$.
Find the derivative of $\sin^{-1}x + \cos^{-1}x$.
Find the derivative of $\tan^{-1}\dfrac{x}{2}$ at $x = 1$.
In the proof for $\sin^{-1}x$, why is $\cos y = +\sqrt{1 - x^2}$?
$\dfrac{d}{dx}\tan^{-1}(e^x) = $