**Q: If y**^{3}=16x^{2}, find dx/dt when x=4 and dy/dt=-1

A:

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

(3y^{2})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:

y^{3}=16(4)^{2}

which gives:

y = cuberoot(256)

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

(3y^{2})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

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