Question

If \( y = e^{\sin(\cosec^{-1}x)} \), then \( \dfrac{dy}{dx} \) is

• A. \( \dfrac{1}{e^x x^2} \)
• B. \( -\dfrac{e^{\frac{1}{x}}}{x^2} \)
• C. \( 0 \)
• D. \( e^{\cos(\cosec^{-1}x)} \)

Hint:

Always simplify inverse trigonometric expressions first before differentiating composite exponential functions.

Answer 1

The Correct Option is B

Step 1: Simplify the given function.
Given \[ y = e^{\sin(\cosec^{-1}x)} \] Using the identity \[ \sin(\cosec^{-1}x) = \frac{1}{x} \] we get \[ y = e^{\frac{1}{x}} \] Step 2: Differentiate with respect to \(x\).
\[ \frac{dy}{dx} = e^{\frac{1}{x}} \cdot \frac{d}{dx}\left(\frac{1}{x}\right) \] Step 3: Compute the derivative.
\[ \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} \] Step 4: Final expression.
\[ \frac{dy}{dx} = -\frac{e^{\frac{1}{x}}}{x^2} \]