Question

What is the derivative of ${\cos }^{-1}\left(x\right)$?

Answer 1

Determine the derivative of ${\cos }^{-1}\left(x\right)$.

In dealing with the derivative of inverse trigonometric functions. We prefer to reorganize and utilize Implicit differentiation since I usually get the inverse derivatives mixed up, and so this way I don't have to memorize them. The chain rule can only be used if you recall the inverse derivatives.

Let
$$\begin{gathered} y={\cos }^{-1}x \\ \Rightarrow \cos y=x \\ \Rightarrow x=cosy \end{gathered}$$

Differentiate Implicitly
$-\sin y\frac{dy}{dx}=1...............\left(i\right)$

By trigonometric identity, we know that

$$\begin{gathered} {\sin }^{2}y+{\cos }^{2}y=1 \\ \Rightarrow {\sin }^{2}y+x^2=1 \\ \Rightarrow {\sin }^{2}y=1-x^2 \\ \Rightarrow \sin y=\sqrt{1-x^2} \end{gathered}$$

By substituting the above value in (i) we obtain:
$$\begin{gathered} -\sqrt{1-x^2}\frac{dy}{dx}=1 \\ \Rightarrow \frac{dy}{dx}=-\frac{1}{\sqrt{1-x^2}} \end{gathered}$$

Hence, the derivative of ${\cos }^{-1}x$ is $"-\frac{1}{\sqrt{1-x^2}}"$.