(1)

\begin{align*} \left(af(x)\right)' & =\lim_{\Delta x\rightarrow0}\frac{af(x+\Delta x)-af(x)}{\Delta x}\\ & =a\lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\\ & =af'(x) \end{align*}

(2)

\begin{align*} \left(f(x)\pm g(x)\right)' & =\lim_{\Delta x\rightarrow0}\frac{\left\{ f(x+\Delta x)\pm g(x+\Delta x)\right\} -\left\{ f(x)\pm g(x)\right\} }{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{\left\{ f(x+\Delta x)-f(x)\right\} \pm\left\{ g(x+\Delta x)-g(x)\right\} }{\Delta x}\\ & \lim_{\Delta x\rightarrow0}\left\{ \frac{f(x+\Delta x)-f(x)}{\Delta x}\pm\frac{g(x+\Delta x)-g(x)}{\Delta x}\right\} \\ & =f'(x)\pm g'(x) \end{align*}

(3)

\begin{align*} \left(af(x)\pm bg(x)\right)' & =\left(af(x)\right)'\pm\left(bg(x)\right)'\\ & =af'(x)\pm bg'(x) \end{align*}

(4)

\begin{align*} \left(f(x)g(x)\right)' & =\lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)g(x+\Delta x)-f(x)g(x)}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)g(x+\Delta x)-f(x)g(x+\Delta x)+f(x)g(x+\Delta x)-f(x)g(x)}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{f(x+\Delta x)-f(x)}{\Delta x}g(x+\Delta x)+f(x)\frac{g(x+\Delta x)-g(x)}{\Delta x}\right)\\ & =f'(x)g(x)+f(x)g'(x) \end{align*}

(5)

\begin{align*} \left(\frac{g(x)}{f(x)}\right)' & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left(\frac{g(x+\Delta x)}{f(x+\Delta x)}-\frac{g(x)}{f(x)}\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta xf(x+\Delta x)f(x)}\left(g(x+\Delta x)f(x)-g(x)f(x+\Delta x)\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta xf(x+\Delta x)f(x)}\left(g(x+\Delta x)f(x)-g(x)f(x)+g(x)f(x)-g(x)f(x+\Delta x)\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{f(x+\Delta x)f(x)}\left(\frac{g(x+\Delta x)-g(x)}{\Delta x}f(x)-g(x)\frac{f(x+\Delta x)-f(x)}{\Delta x}\right)\\ & =\frac{g'(x)f(x)-g(x)f'(x)}{f^{2}(x)} \end{align*}

(6)

\begin{align*} \left(\frac{1}{f(x)}\right)' & =\frac{(1)'f(x)-1f'(x)}{f^{2}(x)}\\ & =-\frac{f'(x)}{f^{2}(x)} \end{align*}