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

Feb 7, 2017

$\frac{d}{\mathrm{dx}} {\cos}^{- 1} \left(x\right) = - \frac{1}{\sqrt{1 - {x}^{2}}}$

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 $y = {\cos}^{- 1} \left(x\right) \iff \cos y = x$

Differentiate Implicitly:

$- \sin y \frac{\mathrm{dy}}{\mathrm{dx}} = 1$ ..... [1]

Using the $\sin \text{/} \cos$ identity;

${\sin}^{2} y + {\cos}^{2} y \equiv 1$
$\therefore {\sin}^{2} y + {x}^{2} = 1$
$\therefore {\sin}^{2} y = 1 - {x}^{2}$
$\therefore \sin y = \sqrt{1 - {x}^{2}}$

Substituting into [1]

$\therefore - \sqrt{1 - {x}^{2}} \frac{\mathrm{dy}}{\mathrm{dx}} = 1$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{\sqrt{1 - {x}^{2}}}$