Final answer:
The differential formula for the change in volume of a sphere is dV = 4π r² dr, where dV is the change in volume, r is the radius of the sphere, and dr is the infinitesimal change in radius.
Step-by-step explanation:
To calculate the differential formula for change in volume of a sphere, we start with the volume formula of a sphere, which is V = 4/3 π r³. When there is a small change in radius, dr, the change in volume, dV, can be found by differentiating the volume formula with respect to r:
dV = d(4/3 π r³)/dr
We apply the power rule for differentiation, which gives us:
dV = 4π r² dr
The differential formula dV = 4π r² dr shows that the change in volume of a sphere (dV) for a given infinitesimal change in radius (dr) is the product of the surface area of the sphere (4π r²) and the change in radius (dr).