Implicit differentiation (Shaum chapter 11, problem 10)

Question

This problem is from the Shaum's outline Calculus. I have started studying implicit differentiation a few weeks ago and I have fared pretty well for all the other problems except this one and another.

I'd like to add that I haven't been through partial derivatives and implicit function theorem yet.

Here we go:

Given $S=πx(x+2y)$ and $V=πx^2y$,I'm to show that $dS/dx=2π(x-y)$ when $V$ is constant, and $dV/dx=-πx(x-y)$ when $S$ is constant.

I just don't know where to start.

Answer 1

If $V=c$ then $S=\pi x(x+\frac {2c} {\pi x^{2}})=\pi x^{2}+\frac {2c} x$. Hence $\frac {dS} {dx}=2\pi x-\frac {2c} {x^{2}}=2 \pi (x-y)$ You can handle the second question in a similar way.

Answer 2

$\textbf{Hint:}$ After you use implicit function theorem on the function that is constant, the function that isn't constant doesn't refer to a partial derivative - i.e.

$$\frac{dS}{dx} = \frac{\partial S}{\partial x} + \frac{\partial S}{\partial y}\cdot\frac{dy}{dx}$$

by chain rule.