The rate of change of the area of a circle with respect to its radius r (in $cm^2/s$), when $r = 6\text{ cm}$ is :

Step 1: Write the formula for the area of a circle

The area of a circle, denoted by $A$, is given by the formula $A = \pi r^2$, where $r$ is the radius of the circle. This formula is fundamental for calculating the area based on its radius.

Step 2: Differentiate the area with respect to the radius

To find the rate of change of the area with respect to the radius, we need to differentiate the area formula $A = \pi r^2$ with respect to $r$. Using the power rule for differentiation, $\frac{d}{dr}(r^2) = 2r$, so $\frac{dA}{dr} = 2\pi r$.

Step 3: Substitute the given radius value

The problem asks for the rate of change when the radius $r = 6$ cm. We substitute this value into the derivative we just calculated. This gives us $2\pi(6)$, which simplifies to $12\pi$.

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