The volume of sphere is given by V = 43πr³. Where r is the radius of the sphere. Find the rate of change of volume with respect to r.

To find the rate of change of the volume of the sphere with respect to the radius, we start with the formula for volume:

V = 43πr³

Next, we need to differentiate V with respect to r. Using the power rule of differentiation, we differentiate:

dV/dr = d/dx(43πr³) = 43π(3r²) = 129πr²

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

dV/dr = 129πr²

This expression tells us how quickly the volume of the sphere increases as the radius increases. For instance, if the radius of the sphere is increased slightly, the change in volume can be approximated using this derivative.