Question

How to differentiate
$\sin^{ - 1} x$

Answer 1

Hey There!!

We can find the derivative as follows:

$\text{Let } y=\sin^{-1}x$

We need to find $\frac{dy}{dx}$


Now,

$y=\sin^{-1}x \\ \\ \\ \implies \sin y = x \\ \\ \\Differentiating \\ \\ \\ \implies \cos y \, \frac{dy}{dx} = 1 \\ \\ \\ \implies \frac{dy}{dx} = \frac{1}{\cos y}$

Now, we know that:

$\sin y = x \\ \\ \\And \, \, \cos^2y=1-\sin^2y \\ \\ \\\implies \cos^2y = 1-x^2 \\ \\ \\ \implies \cos y = \sqrt{1-x^2}$


We can use this in our above sum:
$\frac{dy}{dx}=\frac{1}{\cos y} \\ \\ \\ \implies \boxed{\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}}$


This result should also be remembered. It will make work simpler in many other problems.


Hope it helps
Purva
Brainly Community