Q:  If y³=16x², find dx/dt when x=4 and dy/dt=-1

Answer:

First thing is to take the derivative of the equation implicitly to get:

(3y²)dy/dt = (32x)dx/dt

The goal is to solve for dx/dt… So we need to know y, x, and dy/dt.  We know x and dy/dt (given in the problem).  Use the original equation to solve for y when x is 4:

y³=16(4)²

which gives:

y = cuberoot(256)

So, now we know all that we need to know to solve for dx/dt:

(3y²)dy/dt = (32x)dx/dt

Plug in each variable:

(3(cuberoot256)²)(-1) = (32*4)dx/dt

Simplify:

-3*32*cuberoot(2) = (32*4)dx/dt

(-3/4)cuberoot(2) = dx/dt