multivariable calculus - Calculating partial derivatives

Let f and g be functions of one real variable and define $F(x,y)=f[x+g(y)]$. Find formulas for all the partial derivatives of F of first and second order.

For the first order, I think we have:

$\frac{\partial F}{\partial x}=\frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}$

$\frac{\partial F}{\partial y}=\frac{\partial f}{\partial x}g'(x)+ \frac{\partial f}{\partial y}g'(y)$

Is it correct? What are the second order derivatives?

Thank you

Answer

$f$ is a function of one variable. Therefore the notation $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}$ is problematic (and I suggest you adapt the prime notation in that case). What you have written is not correct.

The correct formulas are: $$\frac{\partial F}{\partial x}(x,y)=f'(x+g(y))$$

$$\frac{\partial F}{\partial y}(x,y)=f'(x+g(y))g'(y)$$ $$\frac{\partial^2 F}{\partial x^2}(x,y)=f''(x+g(y))$$ $$\frac{\partial^2 F}{\partial x \partial y}(x,y)=f''(x+g(y))g'(y)=\frac{\partial^2 F}{\partial y \partial x}(x,y)$$

$$\frac{\partial^2 F}{\partial y^2}(x,y)=f''(x+g(y))g'(y)+f'(x+g(y))g''(y)$$