Question

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

• A. \( e^{x^3} \)
• B. \( 3x^2 e^{x^3} \)
• C. \( 3x^3 e^{x^3} \)
• D. \( 3x^3 e^{x^3} + 3x^2 \)

Hint:

When differentiating composite functions, always apply the chain rule. For derivatives with respect to \( \log x \), remember that the derivative of \( \log x \) is \( \frac{1}{x} \).

Answer 1

The Correct Option is C

We are tasked with finding the derivative of \( e^{x^3} \) with respect to \( \log x \). First, recall the chain rule of differentiation, which states: \[ \frac{d}{dx} \left[ f(g(x)) \right] = f'(g(x)) \cdot g'(x) \] We want to find \( \frac{d}{d(\log x)} e^{x^3} \). We can rewrite this as: \[ \frac{d}{d(\log x)} e^{x^3} = \frac{d}{dx} e^{x^3} \cdot \frac{dx}{d(\log x)} \] Step 1: Differentiating \( e^{x^3} \) with respect to \( x \). By the chain rule, we differentiate \( e^{x^3} \): \[ \frac{d}{dx} e^{x^3} = e^{x^3} \cdot \frac{d}{dx} (x^3) = 3x^2 e^{x^3} \] Step 2: Differentiating \( \log x \) with respect to \( x \). We know that: \[ \frac{d}{dx} (\log x) = \frac{1}{x} \] So, \( \frac{dx}{d(\log x)} = x \). Step 3: Applying the chain rule. Now, applying the chain rule: \[ \frac{d}{d(\log x)} e^{x^3} = 3x^2 e^{x^3} \cdot x = 3x^3 e^{x^3} \] Thus, the derivative of \( e^{x^3} \) with respect to \( \log x \) is \( 3x^3 e^{x^3} \). The correct answer is (C).