We have a function f(x)=log(1/e^x)-ex^3

Evaluate df(x)/dx

f(x)=log1-log(e^x)-ex^3=-x-ex^3

df(x)/dx=-1-3ex^2

Assume the UBC fountain pool is a cylinder with the radius of 3m. Water is added into the pool at a rate of 0.5m^3/min. At what rate (m/min)does the height of the water raise?

Let the height be h.

The volume V=pi3^2h=9pih

Differentiating both sides,

dV/dt=9pi dh/dt

We know that dV/dt=0.5

dh/dt=0.5/9pi=1/18pi

Thus the height changes at a rate of 1/18pi m/min