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

Blaze Jensen

Blaze Jensen

Answered question


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

Answer & Explanation



Beginner2023-03-15Added 7 answers

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 d y d x = 1
Solve for d y d x :
5.) d y d x = 1 cos y
Then, substitution with our original equation yields d y d x in terms of x:
6.) d y d x = 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 ) = 1 - x 2 .
7.) d y d x = 1 1 - x 2
This is a good definition to memorize, along with d d x [ arccos x ] and d d x [ arctan x ] , since they appear quite frequently in differentiation problems.

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