# Thread: Higher order derivatives

1. ## Higher order derivatives

If $\displaystyle u = F(x + g(y))$, show that $\displaystyle u_{x}u_{xy} = u_{y}u_{xx}$.

I have no idea how to go about this, although I am almost certain it involves the chain rule somehow. Any help would be appreciated.

2. Originally Posted by Pinkk
If $\displaystyle u = F(x + g(y))$, show that $\displaystyle u_{x}u_{xy} = u_{y}u_{xx}$.

I have no idea how to go about this, although I am almost certain it involves the chain rule somehow. Any help would be appreciated.
First note that $\displaystyle F$ is only a function of one variable(lets call it $\displaystyle F(t)$) and it is composed with a function of two variables $\displaystyle t=x+g(y)$

Then by the chain rule we have

$\displaystyle \displaystyle u_x=\frac{\partial F}{\partial t}\frac{\partial t}{\partial x}=F_t(x+g(y))$
$\displaystyle \displaystyle u_y=\frac{\partial F}{\partial t}\frac{\partial t}{\partial y}=F_t(x+g(y))\cdot g_y(y)$

This should get you started.

3. Thanks, once I set the argument of $\displaystyle F$ as one variable, the rest followed easily. Thanks again!