Find the partials $\frac{\partial z} {\partial x}$ and $\frac{\partial z}{\partial x}$ with $z$ defined implicitly as a function of $x$ and $y$

Question

Find the partials $\frac{\partial z} {\partial x}$ and $\frac{\partial z}{\partial y}$ with $z$ defined implicitly as a function of $x$ and $y$

The equation is $3x^2z-x^2y^2+2z^3+3yz=5$

I'm confused as to how I should treat the third variable, $z$ when taking the partials. Is this correct?

$\frac{\partial z} {\partial x}= 6xz-2xy^2+2z^3=5$

$\frac{\partial y}{\partial x} = -2x^2y+2z^3+3z=5$

Answer 1

If you are thinking of $z$ as a function of $x$ and $y$, you could write $z=z(x,y)$. Then, when you take a partial derivative with respect to $x$, you must think of $y$ as fixed and study how $z$ changes as you change $x$. In particular, you must use the rules of implicit differentiation:

Starting with: $$3x^2z-x^2y^2+2z^3+3yz=5$$ take the derivative of both sides with respect to $x$ to get $$ 6xz+3x^2\frac{\partial z}{\partial x}-2xy^2+6z^2\frac{\partial z}{\partial x}+3y\frac{\partial z}{\partial x}=0. $$ Now, solve for $\frac{\partial z}{\partial x}$ by factoring and dividing by the coefficient.