Partial derivative implicit differentiation

Question

dz/dx = ?

cos(zy)+zx^2 = (1+y)e^(x-z)

For the left side through implicit differentiation I have found (-sin(zy))(y*(dz/dx))+2xz+(dz/dx)x^2. I am completely unsure how to approach the right side, however.

Answer 1

The problem is quite simple if you use the implicit function theorem.

Consider the implicit function $$F=\cos (y z)+x^2 z-(y+1) e^{x-z}\color{red}{=0}$$ Then $$\frac{\partial F}{\partial x}=2 x z-(y+1) e^{x-z}$$ $$\frac{\partial F}{\partial z}=x^2+(y+1) e^{x-z}-y \sin (y z)$$ $$\frac{\partial z}{\partial x}=-\frac{\frac{\partial F}{\partial x} } {\frac{\partial F}{\partial z} }=\frac{(y+1) e^{x-z}-2 x z}{x^2+(y+1) e^{x-z}-y \sin (y z)}$$