Question

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

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

Hint:

When differentiating inverse trigonometric functions, always apply the chain rule and remember the standard derivative formulas.

Answer 1

The Correct Option is C

Step 1: Use the derivative formula for inverse trigonometric functions.
The derivative of $\cos^{-1}(x)$ is given by: \[ \frac{d}{dx} \cos^{-1}(x) = \frac{-1}{\sqrt{1 - x^2}}. \]

Step 2: Apply this to $\cos^{-1(e^x)$.}
Let $y = \cos^{-1}(e^x)$, then using the chain rule: \[ \frac{dy}{dx} = \frac{-1}{\sqrt{1 - (e^x)^2}} \cdot \frac{d}{dx}(e^x). \] The derivative of $e^x$ is $e^x$. Hence, the derivative becomes: \[ \frac{dy}{dx} = \frac{-e^x}{\sqrt{1 - e^{2x}}}. \]

Step 3: Conclusion.
The correct answer is (C) $\frac{-e^x}{\sqrt{1 - e^{-2x}}}$.