Use the Chain Rule to find $∂z/∂s$ and $∂z/∂t$.
$$ z = \tan(u/v) $$ $$ u = 5s + 7t $$ $$ v = 7s − 5t $$
This is the equation I used to find ∂z/∂s:
$$ \frac{\partial z}{\partial s}=\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial s}+\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial s} $$
Before substituting for $u$ and $v$, I had simplified to:
$$ \frac{\partial z}{\partial s}=\sec^2{\frac{u}{v}}\cdot(5-7\frac{u}{v^2}) $$
However, I can't see how I'll be able to get the equation in terms of $s$. Both $u$ and $v$ are functions on $s$ and $t$.
why dz/ds should have function of s only?you have two independent variable and answer can be function of both of them.for better realize suppose we have a shape in x-y plane and we we know its velocity in cartesian coordinate now you want to obtain its velocity on polar coordinate we expect velocity for example in direction of radius have two Components.i hope you realized my sample. (sorry for my bad english)