Yes, it is long. Is that a problem? Do you only do short problems?
By the chain rule, \(\displaystyle \frac{du}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}+ \frac{\partial f}{\partial z}\frac{dz}{dt}\).
You are, apparently, not given a formula for f so you will have to just leave the partial derivatives.
