# Chain rule for equations of multiple variables

I'm struggling finding some good examples for this function on the internet and I don't really know how to apply the chain rule for functions of this form. Could any of you help me & guide me in the right direction?

EX: Given $F(x,y)=f(x+y,x^2+y^2)$ where $f$ admits continuous partial derivatives of first and second degree. Find: $\frac{\partial^2{F}}{\partial{x^2}},\frac{\partial^2{F}}{\partial{x}\partial{y}}, \frac{\partial^2{F}}{\partial{y^2}}$.

Basically I used notations for $u = x + y$ and $v = x^2 + y^2$ and i found their partial derivatives. What should I do next?

EDIT: So I found this explanation on the internet for a similar problem but I don't understand the part I highlighted in red. How are those two terms products and how is the product rule used there? I've been looking at it for 30 minutes now and it's driving me crazy

• The product is $(\sin\theta)(\frac{\partial f}{\partial \theta})$. – BallBoy Feb 14 '18 at 18:02
• Yes I finally figured that out. Took me waay longer than it should've. Thank you! – Edward B Feb 14 '18 at 18:37

$F(x,y) = f(u,v)\\ \frac {\partial F}{\partial x} = \frac {\partial f}{\partial u}\frac {\partial u}{\partial x}+\frac {\partial f}{\partial v}\frac {\partial v}{\partial x}$
$\frac {\partial^2 F}{\partial x^2} = \frac {\partial}{\partial x}(\frac {\partial f}{\partial u}\frac {\partial u}{\partial x}+\frac {\partial f}{\partial v}\frac {\partial v}{\partial x})$
When you apply the differntial opperator to product such as $\frac {\partial f}{\partial u}\frac {\partial u}{\partial x}$ you will be needing to use both the product rule and the chain rule.
$\frac {\partial^2 F}{\partial x^2} = \frac {\partial^2 f}{\partial u^2}(\frac {\partial u}{\partial x})^2 + \frac {\partial f}{\partial u}\frac {\partial^2 u}{\partial x^2} +\frac {\partial^2 f}{\partial v^2}(\frac {\partial v}{\partial x})^2 + \frac {\partial f}{\partial v}\frac {\partial^2 v}{\partial x^2}$