Question. Given the following information $$z=f(x,y),\quad x=g(u),\quad y=h(u)$$ $$f_x(x,y)=4-x,\quad f_y(x,y)=2y,$$ $$g(1)=4,\quad h(1)=3,\quad g'(1)=2,\quad h'(1)=1.$$ Find $dz/du$ when $u=1$.
I am not sure how to approach this problem. The anti derivative of $x$ and $y$ functions would create $$f(x,y) = y^2 + 4x - \frac{x^2}{2}.$$
So, you have $z$ written as a function of $x$ and $y$, while $x$ and $y$ are each, in turn, functions of $u$. Therefore, $z$ is a function of $u$ (via $x$ and $y$).
To take the derivative of $z$ with respect to $u$, we must consider all ways that $u$ must enter into the formula for $z$. First, it could enter via $x$, and, second, it could enter via $y$. Therefore
$$ \frac{dz}{du}=\frac{\partial z}{\partial x}\frac{dx}{du}+\frac{\partial z}{\partial y}\frac{dy}{du}. $$ Now, what you must do is to plug in $u=1$. You have formulae for all the parts you need, try plugging in yourself.
