problem with chain rule in multivariable calculus

Question

Let $f(x,y)= xy$ and let $y=x^2$. Then is $\frac{\partial f}{\partial x} = x$ or $3x^2$ ?
Please give reasons for your answer.

Answer 1

Neither!

One approach is to consider $f(x,y)= xy$: this defines a surface, we can think of it as the height function $z=xy$.

$\frac{\partial f}{\partial x}$ gives us some information about the tangent plane to a point $(x,y,z)$: it tells us the rate of increase in $f$ as we move in the $x$-direction (whilst keeping the $y$-value unchanged).

For $f(x,y)= xy$ we have $\frac{\partial f}{\partial x}=y$

It doesn't make sense to add the restriction that $y=x^2$. We can't keep $y$ constant while changing $x$ if $y$ depends on $x$.

Alternatively, we can try to consider the curve defined by $f(x,y)= xy$ and $y=x^2$. This is a curve in 3 dimensions that always has coordinates of the form $(x,x^2,x^3)$.

There is no meaning to $\frac{\partial f}{\partial x}$ (because we still can't alter $x$ without changing $y$).

We can, however, look at the change in $z$ from the side, if you like: treating it simply as $z=x^3$. In such a case we can work out $\frac{df}{dx}=3x^2$