Q.Find the rate of change of the area of a circle with respect to its radius $r$ when

The key idea is that the rate of change of area with respect to radius is the derivative $dA/dr$.

Step 1: Area of a circle: $A = \pi r^2$.

Step 2: Differentiate with respect to $r$:

Step 3: Evaluate at the given radii:

• (a) At $r = 3$ cm: $\frac{dA}{dr} = 2\pi (3) = 6\pi$ cm$^2$/cm.
• (b) At $r = 4$ cm: $\frac{dA}{dr} = 2\pi (4) = 8\pi$ cm$^2$/cm.

The question asks for the rate of change of the area of a circle with respect to its radius. That phrase "rate of change" is a direct signal to use a derivative. When one quantity changes as another changes, the instantaneous rate of change is the derivative of the first with respect to the second.

Here, the area $A$ depends on the radius $r$ through the familiar formula $A = \pi r^2$. So the rate of change of $A$ with respect to $r$ is simply $\frac{dA}{dr}$. This derivative tells us how fast the area grows (in square centimeters) for each tiny increase in radius (in centimeters), at a specific value of $r$.

Let’s work through it.

1. Write the relationship. The area of a circle is

2. Differentiate with respect to $r$. Since $\pi$ is a constant,

This is the general formula for the rate of change of area with respect to radius. Notice it is not constant — it grows linearly with $r$. That makes intuitive sense: if you increase the radius of a large circle by 1 cm, you add a much bigger ring of area than if you increase the radius of a tiny circle by the same amount.

3. Evaluate at the given values.

   (a) For $r = 3$ cm:

(b) For $r = 4$ cm:

Method: Finding the Rate of Change of a Geometric Quantity

This method applies whenever a question asks for "the rate of change of [some geometric quantity] with respect to [another quantity]" and gives you a formula linking the two.

Steps

Step 1: Write the formula that relates the two quantities

Identify the quantity you must differentiate (here, an area, volume, or similar) and the variable it depends on. Write down the standard geometric formula connecting them, e.g. $A = \pi r^2$ for the area of a circle in terms of its radius.

Step 2: Differentiate with respect to the variable named in the question

"Rate of change of $Q$ with respect to $x$" always means $\dfrac{dQ}{dx}$. Differentiate the formula from Step 1 using the standard power rule:

This gives you a general expression for the rate — valid for any value of the variable, not yet tied to a specific number.

Step 3: Substitute the given value(s) to get the numerical rate

Only after differentiating, plug in the specific value(s) stated in the question. Each substitution gives the instantaneous rate at that particular value — never substitute before differentiating, or the variable disappears and you'll get zero instead of a rate.

Step 4 (if multiple values are given): Repeat the substitution for each one

Many questions of this type ask for the rate at two or more different values of the variable — simply evaluate the same derivative expression at each value separately; do not re-derive it.

Common Mistakes

Mistake 1: Confusing the derivative with an average rate of change

Since the question gives two specific radii, a student sometimes computes the average rate of change between them, $\dfrac{A(4)-A(3)}{4-3}$, instead of evaluating the derivative $\dfrac{dA}{dr}$ separately at each radius. The phrase "rate of change... when $r=3$" asks for the instantaneous rate at that exact radius, not a change between two states — so $dA/dr=2\pi r$ must be evaluated twice, once at $r=3$ and once at $r=4$.

Mistake 2: Forgetting the units and treating the answer as a plain number

$dA/dr$ has units of cm² per cm (area per unit radius), not cm² or cm. Writing the final answer as just "$6\pi$" or "$8\pi$" without the cm²/cm unit misses that this is a rate, not an area.