Show that a function with implicit dependence is increasing

Question

If a have a function $f(x,y(x))$ and I want to show that it is an increasing function of x, I know that I have to show that the derivative w.r.t. $x$ is positive, but I'm not sure if when I do this I take the partial derivative (i.e. treat $y$ as a constant) or the total derivative (i.e. differentiate the $y(x)$ to get $y'(x)$. Also why would one be correct and not the other? Thank you for your help!!

Answer 1

If you have a function of the form $f(x,y(x))$ this means that the expression of your function will be totally depended from $x$, as $y$ is a function of $x$. Thus, your function is essentially an one variable function $H(x) := f(x,y(x))$. This means that differentiation by $x$ would do the trick, thus checking $H'(x)$ and $H''(x)$.

Treating $y$ as a constant is wrong, since $y$ is a function of $x$, thus it means that it plays a part into the characteristics of the function with respect to $x$.