Circles and Spheres
The function \(A\) with formula \( A(r) = \pi r^2\) returns the area of a circle given its radius. What is a formula for the derivative \(\frac{\mathrm{d}A}{\mathrm{d}r}?\) This formula should look familiar. (This is not a coincidence.)
The function \(V\) with formula \( V(r) = \frac{4}{3}\pi r^3\) returns the volume of a sphere given its radius. What is a formula for the derivative \(\frac{\mathrm{d}V}{\mathrm{d}r}?\) What do you suppose the formula for \(\frac{\mathrm{d}V}{\mathrm{d}r}\) could represent?