Question

(a) Differential coefficient of \( \sin^{-1}(e^{-x}) \) will be:

• A. \( \cos^{-1}(e^{-x}) \)
• B. \( \frac{e^x}{\sqrt{1 - e^{2x}}} \)
• C. \( \frac{1}{\sqrt{e^{2x} - 1}} \)
• D. \( -\frac{1}{\sqrt{e^{2x} - 1}} \)

Hint:

When differentiating inverse trigonometric functions, remember to use the chain rule and simplify carefully.

Answer 1

The Correct Option is D

Tofindthederivativeof\(\sin^{-1}(e^{-x})\): \[ \text{Let}y=\sin^{-1}(e^{-x}). \] Differentiatingbothsideswithrespectto\(x\): \[ \frac{dy}{dx}=\frac{1}{\sqrt{1-(e^{-x})^2}}\cdot\frac{d}{dx}(e^{-x}). \] \[ \frac{dy}{dx}=\frac{1}{\sqrt{1-e^{-2x}}}\cdot(-e^{-x}). \] Simplifying: \[ \frac{dy}{dx}=-\frac{1}{\sqrt{e^{2x}-1}}. \]