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To solve the problem, we are asked to find the rate of change of the volume \( V \) of a cylinder with respect to its radius \( r \), given that the height \( h \) of the cylinder is equal to its radius.

### Step 1: Formula for the Volume of a Cylinder
The volume \( V \) of a cylinder is given by the formula:

\[
V = \pi r^2 h
\]

Since the problem states that the height is equal to the radius, we can substitute \( h = r \) into the equation:

\[
V = \pi r^2 \cdot r = \pi r^3
\]

### Step 2: Differentiate Volume with Respect to Radius
We now need to find the derivative of \( V \) with respect to \( r \). This means we will differentiate \( V = \pi r^3 \) using the power rule:

\[
\frac{dV}{dr} = \frac{d}{dr} (\pi r^3) = 3\pi r^2
\]

Thus, the rate of change of volume with respect to radius is:

\[
\frac{dV}{dr} = 3\pi r^2
\]

This is the required expression for the derivative.

Would you like any further clarifications or details on this problem?

### Related Questions:
1. How would the derivative change if the height of the cylinder was a constant value instead of being equal to the radius?
2. What is the physical interpretation of the derivative \( \frac{dV}{dr} \) in this context?
3. How does the volume change if the radius is doubled?
4. Can you generalize the volume formula for a cone with a height equal to its radius?
5. What would happen to the volume formula if the height was proportional to the square of the radius?

### Tip:
When solving related rates problems, it is essential to correctly identify the variables that are dependent and independent to apply the chain rule properly.

A cylinder has a height equal to its radius. Find the rate of change of volume V with respect to radius.

Learn how to find the rate of change of the volume of a cylinder when its height is equal to its radius. This problem explores key calculus concepts, including the power rule for derivatives and the volume formula for cylinders, suitable for advanced high school students.

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