What is the derivative of y = arcsin ( x ) ?

Question

What is the derivative of       y    =          arcsin              (        x        )            ?

dhopeefreshnwn · Accepted Answer

This identity can be proven easily by applying sin to both sides of the original equation:
1.)

y = \arcsin x

2.)

\sin y = \sin (\arcsin x)

3.)

\sin y = x

We continue by using implicit differentiation, keeping in mind to use the chain rule on sin y:
4.)

\cos y \frac{d y}{d x} = 1

Solve for

\frac{d y}{d x}

:
5.)

\frac{d y}{d x} = \frac{1}{\cos y}

Then, substitution with our original equation yields

\frac{d y}{d x}

in terms of x:
6.)

\frac{d y}{d x} = \frac{1}{\cos (\arcsin x)}

This may not appear to be a good idea at first, but it can be simplified if the identity is remembered.

\sin (\arccos x) = \cos (\arcsin x) = \sqrt{1 - x^{2}}

.
7.)

\frac{d y}{d x} = \frac{1}{\sqrt{1 - x^{2}}}

This is a good definition to memorize, along with

\frac{d}{d x} [\arccos x]

and

\frac{d}{d x} [\arctan x]

, since they appear quite frequently in differentiation problems.

What is the derivative of y=arcsin(x)?