# Thread: Find partial derivative...

1. ## Find partial derivative...

Find the partial derivatives of $F(x,y) = f( g(x) k(y), h(x) + 2k(y) )$.

Partial derivatives are easy for me when the function is written g(x,y) = x^2 - sin(y) or something like that, but when it's written as a composition of functions, I start to get confused. Could someone please help me with this?

2. just use the chain rule and we make the following definitions

$p(x,y):=g(x)k(y)$
$q(x,y):=g(x)+2k(y)$

$\frac{\partial F}{\partial x}=\frac{\partial f}{\partial p}\frac{\partial p}{\partial g}\frac{\partial g}{\partial x}+\frac{\partial f}{\partial q}\frac{\partial q}{\partial g}\frac{\partial g}{\partial x}$

and similarly for y, remembering that y is in the functions k