Exercise 12

Find the indicated derivative. In each case, state the version of the Chain Rule that you are using.

1. $\frac{df}{dt}$, if $f(x,y) = 2x^2y$, $x=\cos(t)$, and $y=\ln(t)$.

2. $\frac{\partial f}{\partial w}$, if $f(x,y) = 2x^2y$, $x=w+z^2$, and $y=\frac{2z+1}{w}$

3. $\frac{\partial f}{\partial v}$, if $f(x,y,z) = 2x^2y+z^3$, $x=u-v+2w$, $y=w2^v-u^3$, and $z = u^2-v$