Quick question on how to handle negative fraction exponents when differentiating:
I have this problem to differentiate.
$$x^{2/3} + y^{2/3} = 1$$
So my textbook and I both did the first thing the same way:
$$\frac23x^{-1/3} + \frac23y^{-1/3} \frac{dy}{dx} = 0$$
But here is where I got a bit lost.. I simplified the negative exponent and came up with:
$$\frac{dy}{dx} = -\frac{2}{3x^3} \bigg/ \frac{3}{3y^3} \implies -\frac{y^3}{x^3}$$
My textbook gives this instead:
$$-\left(\frac{y}{x}\right)^{1/3}$$
Can anyone explain how that happened? Unfortunately I'm taking this college subject online and the college forum where we communicate with our teachers isn't very active.
$x^{-1/3} = \frac{1}{x^{1/3}}$, not $\frac{1}{x^3}$.