I'm watching the first MIT OCW 18.01 lecture. In this lecture, the lecturer gives a treatment of finding the derivative for $f(x) = \frac{1}{x}$. He presents this interstitial step $$\frac{\Delta f}{\Delta x} = \frac{\frac{1}{x_0+\Delta x} - \frac{1}{x_0}}{\Delta x}$$ and simplifies it to $$\frac{\Delta f}{\Delta x} = \frac{1}{\Delta x}\left(\frac{x_0 - \left(x_0 + \Delta x\right)}{\left(x_0 + \Delta x\right)x_0}\right)$$.
He explains the simplification as factoring out the $\frac{1}{\Delta x}$ from the denominator and then multiplying the resulting denominators $x_0 + \Delta x$ and $x_0$ to obtain the denominator $\frac{x_0 - \left(x_0 + \Delta x\right)}{\left(x_0 + \Delta x\right)x_0}$ and then "figuring out what the numerator had to be". It's not clear to me how the numerator in this case would be anything other than $-1$; indeed, after some simplification, he ends up with $\frac{\Delta f}{\Delta x} = \frac{-1}{x^2}$.
What are the steps involved in simplifying the numerator in this expression, and what am I missing here? To try to be more clear, I understand that ultimately the numerator cancels to be $-1$. I understand the power rule and how to use it to short-circuit round the step by step procedure. What I am not clear on is how, in the intermediate step, the numerator is an expression involving $x_0$ and $\Delta x$. How did this come up? Where does it come from?
I am not completely clear on what you are asking, but the numerator $x_0-(x_0+\Delta x)$ comes from this: $$\frac{1}{x_0+\Delta x} - \frac{1}{x_0}=\frac{x_0}{x_0(x_0+\Delta x)}-\frac{x_0+\Delta x}{x_0(x_0+\Delta x)}=\frac{x_0-(x_0+\Delta x)}{x_0(x_0+\Delta x)}$$