$$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_1-x_2}=\frac{f(a)-f(b)}{a-b}$$
$$f'(x_0)=\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=\lim_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}$$
\begin{align*} f'(1)&=\lim_{h\to 0}\frac{f(1+h)-f(1)}{h}\\ &=\lim_{h\to 0}\frac{(1+h)^3-1^3}{h}\\ &=\lim_{h\to 0}\frac{(1+h)\cdot(1+h)^2-1}{h}\\ &=\lim_{h\to 0}\frac{(1+h)\cdot(1^2+2h + h^2)-1}{h}\\ &=\lim_{h\to 0}\frac{1+3h+3h^2+h^3-1}{h}\\ &=\lim_{h\to 0}\frac{3h+3h^2+h^3}{h}\\ &=\lim_{h\to 0}\frac{h(3+3h+h^2)}{h}\\ &=\lim_{h\to 0}(3+3h+h^2)\\ &=3+3\cdot 0+0^2\\ &=3 \end{align*}