Differentiation of composed function.

Question

Using the rule of differentiation composed function calculate the first order partial derivatives of $x$ and $y$: $$ z = f(u,v,w) = \arcsin \frac{u}{v+w}$$ $$u= e^\frac{x}{y}, v= x^2 + y^2, w =2xy$$ Please help me.

Answer 1

First trick: Since $v$ and $w$ only appear added together, and since they're both polynomials, it makes sense to combine them - let $g(x, y) = v + w = x^2 + y^2 + 2xy = (x + y)^2$.

So now, $z = \arcsin{\frac{u(x, y)}{g(x, y)}}$. It looks nicer already.

Next, you need to calculate the partial derivatives of $u$ and $g$ with respect to $x$ and $y$. $\frac{\partial u}{\partial x} = \frac{\partial}{\partial x}e^\frac{x}{y} = \frac{1}{y}e^\frac{x}{y}$ and so forth.

Then, find the partial derivatives of $z$ in terms of $u, g$ and their respective partials.

$\begin{eqnarray} \frac{\partial z}{\partial x} & = & \frac{\partial}{\partial x} \arcsin \frac{u}{g} \\ & = & \frac{-1}{\sqrt{1-\left(\frac{u}{g}\right)^2}}\frac{\partial}{\partial x} \frac{u}{g} \\ & = & \frac{-\left | g \right |}{\sqrt{g^2 - u^2}}\frac{\frac{\partial u}{\partial x}g - u\frac{\partial g}{\partial x}}{g^2} \end{eqnarray}$

and similarly for y.

Next, substitute in everything you know.

Finally, don't trust me to have gotten a single step in those derivatives right and check the results yourself.