What is the derivative of $sin^-1(x)$ ?

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Answer 1 · 2016-04-15T13:35:03

$1/sqrt(1-x^2)$

Let $y=sin^-1x$ ,
so $siny=x$ and $-pi/2 <= y <= pi/2$ (by the definition of inverse sine).

Now differentiate implicitly:

$cosy dy/dx = 1$ , so

$dy/dx = 1/cosy$ .

Because $-pi/2 <= y <= pi/2$ , we know that $cosy$ is positive.

So we get:

$dy/dx = 1/sqrt(1-sin^2y) = 1/sqrt(1-x^2)$ . (Recall from above $siny=x$ .)

What is the derivative of sin^-1(x)sin^-1(x)?