Question

The derivative of \( e^{x^{3}} \) with respect to \( \log x \) is:

• A. $3x^{2}e^{x^{3}}$
• B. $3x^{3}e^{x^{3}}$
• C. $\frac{e^{x^{3}}}{x}$
• D. $3x e^{x^{3}}$

Hint:

Always differentiate both functions with respect to $x$ first when doing relative derivatives.

Answer 1

The Correct Option is B

Step 1: Concept
$\frac{du}{dv} = \frac{du/dx}{dv/dx}$.
Step 2: Analysis
Let $u = e^{x^{3}}$ and $v = \log x$.
$du/dx = e^{x^{3}} \cdot 3x^{2}$ and $dv/dx = 1/x$.
Step 3: Conclusion
$\frac{du}{dv} = \frac{3x^{2}e^{x^{3}}}{1/x} = 3x^{3}e^{x^{3}}$.
Final Answer: (B)