Partial Derivatives - Chain Rule Multiple Variable

Question

Suppose that $z$ and $w$ are differentiable functions of $x$ and $y$ satisfying the equations $$xw^3-z^3+xy^2=1$$ and $$yw^2-xz^2+x^3=7$$ find $\frac{\partial z}{\partial x}$ when $(x,y,z,w) = (2,1,-1,-1)$

Answer 1

We have

• $xw^3-z^3+xy^2=1\implies w^3+3xw^2w_x-3z^2z_x+y^2=0\\\implies -1+6w_x-3z_x+1=0\implies z_x=2w_x$
• $yw^2-xz^2+x^3=7\implies 2yww_x-z^2-2xzz_x+3x^2=0\\\implies -2w_x-1+4z_x+12=0\implies w_x=2z_x+\frac{11}2$

substituting the latter in the first

$$z_x=2\left(2z_x+\frac{11}2\right)=4z_x+11\implies z_x=-\frac{11}3$$