Proofs of various derivative rules
Derivative of a constant
$$\begin{split}
\textrm{let }f(x) &= c
\\\frac{d}{dx}f(x) &= \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{c-c}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{0}{\Delta x}
\\&= 0
\end{split}$$
Constant multiple rule
$$\begin{split}
\frac{d}{dx}cf(x) &= \lim_{\Delta x\to0}\frac{cf(x+\Delta x)-cf(x)}{\Delta x}
\\&= c\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
\\&= cf'(x)
\end{split}$$
Sum rule
$$\begin{split}
\frac{d}{dx}(f(x)+g(x)) &= \lim_{\Delta x\to0}\frac{(f(x+\Delta x)+g(x+\Delta x))-(f(x)+g(x))}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{f(x+\Delta x)+g(x+\Delta x)-f(x)-g(x)}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)+g(x+\Delta x)-g(x)}{\Delta x}
\\&= \lim_{\Delta x\to0}\left(\frac{f(x+\Delta x)-f(x)}{\Delta x}+\frac{g(x+\Delta x)-g(x)}{\Delta x}\right)
\\&= \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}+\lim_{\Delta x\to0}\frac{g(x+\Delta x)-g(x)}{\Delta x}
\\&=f'(x)+g'(x)
\end{split}$$
Product rule
$$\begin{split}
\frac{d}{dx}f(x)g(x) &= \lim_{\Delta x\to0}\frac{f(x+\Delta x)g(x+\Delta x)-f(x)g(x)}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{f(x+\Delta x)g(x+\Delta x)-f(x+\Delta x)g(x)+f(x+\Delta x)g(x)-f(x)g(x)}{\Delta x}
\\&= \lim_{\Delta x\to0}\left(\frac{f(x+\Delta x)g(x+\Delta x)-f(x+\Delta x)g(x)}{\Delta x}+\frac{f(x+\Delta x)g(x)-f(x)g(x)}{\Delta x}\right)
\\&= \lim_{\Delta x\to0}\frac{f(x+\Delta x)g(x+\Delta x)-f(x+\Delta x)g(x)}{\Delta x}+\lim_{\Delta x\to0}\frac{f(x+\Delta x)g(x)-f(x)g(x)}{\Delta x}
\\&= \lim_{\Delta x\to0}f(x+\Delta x)\frac{g(x+\Delta x)-g(x)}{\Delta x}+\lim_{\Delta x\to0}g(x)\frac{f(x+\Delta x)-f(x)}{\Delta x}
\\&= \lim_{\Delta x\to0}\left(f(x+\Delta x)\right)\lim_{\Delta x\to0}\left(\frac{g(x+\Delta x)-g(x)}{\Delta x}\right)+\lim_{\Delta x\to0}\left(g(x)\right)\lim_{\Delta x\to0}\left(\frac{f(x+\Delta x)-f(x)}{\Delta x}\right)
\\&= f(x)g'(x)+g(x)f'(x)
\end{split}$$
Quotient rule
$$\begin{split}
\frac{d}{dx}\frac{f(x)}{g(x)} &= \lim_{\Delta x\to0}\frac{\frac{f(x+\Delta x)}{g(x+\Delta x)}-\frac{f(x)}{g(x)}}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{1}{\Delta x}\left(\frac{f(x+\Delta x)}{g(x+\Delta x)}-\frac{f(x)}{g(x)}\right)
\\&= \lim_{\Delta x\to0}\frac{1}{\Delta x}\left(\frac{f(x+\Delta x)g(x)}{g(x+\Delta x)g(x)}-\frac{f(x)g(x+\Delta x)}{g(x+\Delta x)g(x)}\right)
\\&= \lim_{\Delta x\to0}\frac{1}{\Delta x}\left(\frac{f(x+\Delta x)g(x)-f(x)g(x+\Delta x)}{g(x+\Delta x)g(x)}\right)
\\&= \lim_{\Delta x\to0}\frac{1}{\Delta x}\left(\frac{f(x+\Delta x)g(x)-f(x)g(x)+f(x)g(x)-f(x)g(x+\Delta x)}{g(x+\Delta x)g(x)}\right)
\\&= \lim_{\Delta x\to0}\frac{1}{g(x+\Delta x)g(x)}\left(\frac{f(x+\Delta x)g(x)-f(x)g(x)+f(x)g(x)-f(x)g(x+\Delta x)}{\Delta x}\right)
\\&= \lim_{\Delta x\to0}\frac{1}{g(x+\Delta x)g(x)}\left(g(x)\frac{f(x+\Delta x)-f(x)}{\Delta x}-f(x)\frac{g(x+\Delta x)-g(x)}{\Delta x}\right)
\\&= \lim_{\Delta x\to0}\frac{1}{g(x+\Delta x)g(x)}\left(\lim_{\Delta x\to0}g(x)\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}-\lim_{\Delta x\to0}f(x)\lim_{\Delta x\to0}\frac{g(x+\Delta x)-g(x)}{\Delta x}\right)
\\&= \frac{1}{g(x)g(x)}\left(g(x)f'(x)-f(x)g'(x)\right)
\\&= \frac{g(x)f'(x)-f(x)g'(x)}{\left(g(x)\right)^2}
\end{split}$$
Power rule
$$\begin{split}
\frac{d}{dx}x^n &= \lim_{\Delta x\to0}\frac{\left(x+\Delta x\right)^n-x^n}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{\sum^{n}_{i=0}\binom{n}{i}x^i\Delta x^{n-i}-x^n}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{x^n+\Delta x\binom{n}{n-1}x^{n-1}+\Delta x^2\sum^{n-2}_{i=0}\binom{n}{i}x^i\Delta x^{n-2-i}-x^n}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{\Delta x\binom{n}{n-1}x^{n-1}+\Delta x^2\sum^{n-2}_{i=0}\binom{n}{i}x^i\Delta x^{n-2-i}}{\Delta x}
\\&= \lim_{\Delta x\to0}\frac{\Delta x\left(\binom{n}{n-1}x^{n-1}+\Delta x\sum^{n-2}_{i=0}\binom{n}{i}x^i\Delta x^{n-2-i}\right)}{\Delta x}
\\&= \lim_{\Delta x\to0}\binom{n}{n-1}x^{n-1}+\Delta x\sum^{n-2}_{i=0}\binom{n}{i}x^i\Delta x^{n-2-i}
\\&= \binom{n}{n-1}x^{n-1}
\\&= \frac{n!}{\left(n-1\right)!\left(n-\left(n-1\right)\right)!}x^{n-1}
\\&= \frac{n!}{\left(n-1\right)!1!}x^{n-1}
\\&= \frac{n!}{\left(n-1\right)!}x^{n-1}
\\&= nx^{n-1}
\end{split}$$
Chain rule
$$\begin{split}
\frac{d}{dx}f(g(x))) &= \lim_{\Delta x\to0}\left(\frac{f(g(x+\Delta x))-f(g(x))}{\Delta x}\right)
\\&= \lim_{\Delta x\to0}\left(\frac{f(g(x+\Delta x))-f(g(x))}{\Delta x}\cdot\frac{g(x+\Delta x)-g(x)}{g(x+\Delta x)-g(x)}\right)
\\&= \lim_{\Delta x\to0}\left(\frac{f(g(x+\Delta x))-f(g(x))}{g(x+\Delta x)-g(x)}\cdot\frac{g(x+\Delta x)-g(x)}{\Delta x}\right)
\\&= \lim_{\Delta x\to0}\left(\frac{f(g(x+\Delta x))-f(g(x))}{g(x+\Delta x)-g(x)}\right)\cdot\lim_{\Delta x\to0}\left(\frac{g(x+\Delta x)-g(x)}{\Delta x}\right)
\\&= \lim_{\Delta x\to0}\left(\frac{f(g(x+\Delta x))-f(g(x))}{g(x+\Delta x)-g(x)}\right)\cdot g'(x)
\\&= \lim_{k\to0}\left(\frac{f(u+k)-f(u)}{k}\right)g'(x)
\\&= f'(u)g'(x)
\\&= f'(g(x))g'(x)
\end{split}$$
$$\begin{split}
&\textrm{let }k=g(x+\Delta x)-g(x).
\\&\textrm{let }u=g(x).
\\\\&\textrm{Properties:}
\\&\hspace{1.5em}g(x+\Delta x)=u+k.
\\&\hspace{1.5em}k\to0\textrm{ as } \Delta x\to0.
\end{split}$$