Find the second derivative of the volume of a cube with respect to the length of a side
v = x^3
dv/dx = 2 x^2
d^2 v/dx^2 = 4 x
v = x^3
dv/dx = 3 x^2
d^2 v/dx^2 = 6 x
To find the second derivative of the volume of a cube with respect to the length of a side, we need to follow these steps:
1. Express the volume of the cube in terms of the length of a side. Since a cube has all sides equal, we can denote the length of a side as "s". Therefore, the volume of the cube is given by V = s^3.
2. Differentiate the volume function V with respect to "s" to find the first derivative. In this case, dV/ds = 3s^2. This represents the rate at which the volume changes as the length of a side changes.
3. Differentiate the first derivative dV/ds with respect to "s" to find the second derivative. In this case, d^2V/ds^2 = d/ds (3s^2) = 6s. This represents the rate at which the rate of change of volume changes as the length of a side changes.
Therefore, the second derivative of the volume of a cube with respect to the length of a side is 6s.