Total differential $\mathrm{d}h$, finding function $h(x,y)$

Question

I have the total differential of ${\mathrm d}h=(4x+4y){\mathrm d}x+(4x+8y){\mathrm d}y$.

How do I determine the function $h(x,y)$ that gives total differential $\mathrm{d}h$?

Answer 1

HINT

$$ {\rm d}h = \frac{\partial h}{\partial x}{\rm d}x + \frac{\partial h}{\partial y}{\rm d}y $$

When you compare that with your original equation, you can conclude that

$$ \frac{\partial h}{\partial x} = 4x + 4y ~~~~\mbox{and}~~~~~ \frac{\partial h}{\partial y} = 4x + 8y $$