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

1 Answer
Feb 7, 2017

d/dxcos^(-1)(x) = -1/sqrt(1 -x^2)

Explanation:

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you can use the chain rule.

Let y=cos^(-1)(x) <=> cosy=x

Differentiate Implicitly:

-sinydy/dx = 1 ..... [1]

Using the sin"/"cos identity;

sin^2y+cos^2y -= 1
:. sin^2y+x^2 = 1
:. sin^2y = 1 -x^2
:. siny = sqrt(1 -x^2)

Substituting into [1]

:. -sqrt(1 -x^2)dy/dx=1
:. dy/dx = -1/sqrt(1 -x^2)