Question

how does the first equation become the second? (calculus)

Answer 1

\[-[p(x+\Delta(x))-p(x)] \]
\[\frac{ -dp }{ dx}\Delta(x)\]

Answer 2

Let's revisit the `Limit Definition of a Derivative`.\[\large \lim_{\Delta x \rightarrow 0}\frac{p(x+\Delta x)-p(x)}{\Delta x} \qquad = \qquad \frac{dp}{dx}\]

Answer 3

If we multiply both sides by −Δx, I think we get there.
\[\large -{\Delta x}\left(\lim_{\Delta x \rightarrow 0}\frac{p(x+\Delta x)-p(x)}{\Delta x} \right)\qquad = \qquad -\Delta x\left(\frac{dp}{dx}\right)\]