I'm trying to find the derivative of the following using the limit definition of a derivative: $$f(x)=x^{2/3}.$$
I know that the derivative of $f(x)$ is $\frac23x^{-1/3}$ by the power rule, but I can't figure out how to do it with the limit definition. I've tried multiplying by a conjugate and can't get it that way. The professor said to use the difference of cubes formula to solve it, but I can't figure out how that plays into this question. The question is due 9/6/23, so any answers before that are greatly welcomed.
This might a bit general and lengthy (compared to @Feng), but you could start off proving the power rule and extending it to rational numbers with a proof I found.
Start by differentiating $y=x^{m/n}$ on both sides with respect to $x$. $$LHS=\frac{d}{dx}y^n=\frac{d}{dy}y^n\frac{dy}{dx}=ny^{n-1}\frac{dy}{dx}$$ $$RHS=mx^{m-1}$$ $$\therefore ny^{n-1}\frac{dy}{dx} = mx^{m-1}$$ $$\therefore \frac{dy}{dx} = \frac{mx^{m-1}}{ny^{n-1}} = \frac{mx^{m-1}}{n(x^{\frac{m}{n}})^{n-1}}=\frac{m}{n}x^{\frac{m}{n}-1}$$ This proves that rational exponents work for the power rule too (I know this isn't actually a proof, but you can write it out formally for your work).
I did see @Feng's comment while writing this, and his method is much better than mine. I just put this here for reference.