Question
Differentiate $\cos^{–1}(e^x)$ w.r.t. $x.$
✓
Answer
Let $y = \cos^{–1} (e^x)$.
Using the chain rule, we have
$\frac{d y}{d x}=\frac{-1}{\sqrt{1-\left(e^{x}\right)^{2}}} \cdot \frac{d}{d x}\left(e^{x}\right)=\frac{-e^{x}}{\sqrt{1-e^{2 x}}}$
Using the chain rule, we have
$\frac{d y}{d x}=\frac{-1}{\sqrt{1-\left(e^{x}\right)^{2}}} \cdot \frac{d}{d x}\left(e^{x}\right)=\frac{-e^{x}}{\sqrt{1-e^{2 x}}}$
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