Let $w(x, y, z) = \sqrt{x^2 + y^2 + z^2}$ where $x = -4re^t$, $y = 7te^r$ & $z = e^{rt}$.
Calculate $\frac{\partial w}{\partial r}$ & $\frac{\partial w}{\partial t}$ by first finding $\frac{\partial x}{\partial r}$, $\frac{\partial y}{\partial r}$, $\frac{\partial z}{\partial r}$, $\frac{\partial x}{\partial t}$, $\frac{\partial y}{\partial t}$ & $\frac{\partial z}{\partial t}$ and using the chain
rule.
$\frac{\partial w}{\partial r} = $
$\frac{\partial w}{\partial t} = $
