Question

Derivative of \( x^2 \) with respect to \( x^3 \), is:

• A. \( \frac{2}{3x} \)
• B. \( \frac{3x}{2} \)
• C. \( \frac{2x}{3} \)
• D. \( 6x^5 \)

Hint:

When differentiating with respect to a different variable, apply the chain rule and adjust for the powers of the variables.

Answer 1

The Correct Option is A

We are asked to differentiate \( x^2 \) with respect to \( x^3 \), so we need to use the chain rule. \[ \frac{d}{dx} \left( \frac{x^2}{x^3} \right) = \frac{d}{dx} \left( x^{2 - 3} \right) = \frac{d}{dx} \left( x^{-1} \right) \] \[ \frac{d}{dx} \left( x^{-1} \right) = -x^{-2} = \frac{2}{3x} \] 
Step 2: Verify the options
The correct derivative is \( \frac{2}{3x} \), matching option (A).