Sodium chlorate crystals are easy to grow in the shape of cu

Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain its meaning (be specific). Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube.

Solution

volume of cube = x^3 a) dV/dx = 3x^2 at x = 3 dV/dx = 3*9 = 27mm^2 b) dV/dc = 3x^2 = 27 S = 6x^2 = 6*9 = 54 hence dV/dx = S/2