Assume a cube with all sides equal to x. Let the change in x with respect to t; x(t) = 2t + 1. What is the formula for the rate of change of the volume of the cube?

v = x^3, so

dv/dt = 3x^2 dx/dt
= 6(2t+1)^2

To find the formula for the rate of change of the volume of the cube, we first need to express the volume as a function of x.

Since all sides of the cube are equal to x, the volume V is given by V = x^3, as the cube's volume is the side length cubed.

Now, we are given that x is a function of t: x(t) = 2t + 1. We can substitute this expression into the formula for volume:

V = (2t + 1)^3

Next, we want to find the rate of change of the volume with respect to t, denoted as dV/dt or V'. To achieve this, we can differentiate V with respect to t:

V' = dV/dt = d/dt (2t + 1)^3

To differentiate (2t + 1)^3, we can apply the chain rule. Let's denote u = 2t + 1:

V' = d/dt (u^3)

Using the chain rule, the derivative of u^3 with respect to t is:

V' = 3u^2 * du/dt

Now, we need to find du/dt by differentiating u = 2t + 1 with respect to t:

du/dt = d/dt (2t + 1) = 2

Substituting the values back into V', we get:

V' = 3u^2 * du/dt = 3(2t + 1)^2 * 2

Simplifying further, the formula for the rate of change of the volume of the cube is:

V' = 6(2t + 1)^2