Question

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

Answer 1

`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)`