Find partial derivatives of $f(x,y) = F(f_1(x,y), f_2(x,y), f_3(x,y))$ in terms of the partial derivatives of $F.$

Question

We have $f(x,y)= F(\cos(xy), y^2\log(1+x^2),\arctan(x^4+y^2)).$

I think that $f_x = F_x \cdot y\sin(xy)+ F_y\cdot \frac{2xy^2}{1+x^2}+F_z\cdot \frac{4x^3}{1+(x^4+y^2)^2}$

Is this correct?

Answer 1

We have that

$$f_x=\frac{\partial F}{\partial f_1}\frac{\partial f_1}{\partial x}+\frac{\partial F}{\partial f_2}\frac{\partial f_2}{\partial x}+\frac{\partial F}{\partial f_3}\frac{\partial f_3}{\partial x}$$

note that

$$\frac{\partial (\cos (xy))}{\partial x}=-y\sin(xy)$$

maybe is a little confusing use $F_x,F_y,F_z$ for the partial derivative of $F$, we could use $F_1,F_2,F_3$ or $F_u,F_v,F_w$.