Question

Differentiate the following w.r.t. $x$: $\cos e{c}^{-1}\left(e^{-x}\right)$

Answer 1

Solution:

The differentiation of the given function is computed by using the chain rule.

Let $y=\cos e{c}^{-1}u\mathrm{and}u=e^{-x}$

$$\begin{gathered} \frac{dy}{dx}=\frac{dy}{du}·\frac{du}{dx} \\ =\frac{d}{du}(\cos e{c}^{-1}u)·\frac{d}{dx}\left(e^{-x}\right) \\ =-\frac{1}{u\sqrt{u^2-1}}·(-e^{-x}) \\ =\frac{e^{-x}}{e^{-x}\sqrt{e^{-2x}-1}}\left[\because u=e^{-x}\right] \\ =\frac{1}{\sqrt{e^{-2x}-1}}or\frac{e^x}{\sqrt{1-e^{2x}}} \end{gathered}$$

Hence, the required answer is $\frac{1}{\sqrt{e^{-2x}-1}}or\frac{e^x}{\sqrt{1-e^{2x}}}$.