Question

Multiple Choice Find the instantaneous rate of change of the volume of a cube with respect to a side length \(x .\) $$\begin{array}{llll}{\text { (A) } x} & {\text { (B) } 3 x} & {\text { (C) } 6 x} & {\text { (D) } 3 x^{2}} & {\text { (E) } x^{3}}\end{array}$$

Short answer

The correct answer is (D), the rate of change of the volume of a cube with respect to its side length is \( 3x^2 \).

Step-by-step solution

Understanding the Problem

From the problem, we gather that we are looking for the rate of change of the volume of a cube with respect to its side length \( x \). We know that the volume of a cube is given by \( V = x^3 \), where \( V \) is the volume and \( x \) is the side length of the cube.

Applying the Power Rule for Differentiation

To find the rate of change (or derivative) of the volume \( V \) with respect to the side length \( x \), we can use the power rule for differentiation. This rule states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \). For our case, \( n=3 \), so taking the derivative we get \( \frac{dV}{dx} = 3x^{3-1} \). Simplifying this gives \( \frac{dV}{dx} = 3x^2 \). Therefore, the rate of change of the volume with respect to the side length is \( 3x^2 \).