I am pretty new to multivariable calculus, I know how to find $f_x$, $f_{xx}$, $f_{xy}$ etc. but that's about all.
I want to solve this question:
Find the value of $\frac{∂z}{∂x}$ at the point (1, 1, 1) if the equation $2xy + z^3x − yz = 2$ defines $z$ as a function of the two independent variables $x$ and $y$.
Edit:
Thanks to all the comments, I went ahead and tried to answer the question. $$2xy + z^3x − yz = 2$$
$$2y+z^3+3z^2x\frac{∂z}{∂x}-y\frac{∂z}{∂x}=0$$
$$\frac{∂z}{∂x}(3z^2x-y)=-2y-z^3$$
$$\frac{∂z}{∂x}=\frac{-2y-z^3}{3z^2x-y}$$
at $(1, 1, 1)$:
$$\frac{∂z}{∂x}=\frac{-2(1)-(1)^3}{3(1)^2(1)-(1)}=-\frac{3}{2}$$
Is this correct?
Thanks!