How do I solve a problem that involves finding the derivative of of x respect to y?

Question

I have done problems where I have to find the derivative of $y$ with respect to $x$, but not the other way around. Do I do the problem just like the other problems? I'm not sure where to start.

Let $y=x^7+2x-5$. Calculate $\dfrac{dx}{dy}\bigg\vert_{y=-2}$ the derivative of $x$ with respect to $y$ when $y=-2$.

Answer 1

HINT

Take the derivative of your equation with respect to $y$, you get $$ 1 = 7x^6 \frac{dx}{dy} + 2 \frac{dx}{dy}, $$ can you complete this now?

This is called implicit differentiation.

Answer 2

Note that at $y=-2$ we have $x=1$

$$y=x^7+2x-5$$

$$ 1= 7x^6 x' +2x'$$

$$ x'(2+7x^6)=1$$

$$x'(9)=1 \implies x'=1/9$$

Answer 3

Also recognize that $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$. So you could just solve for $\frac{dy}{dx}$ and then flip it. Note that this does not work for higher-order derivatives for reasons stated here.