Question

Find the derivation of the following function from first principle.
(1) $x^3-27$
(2) $(x-1)(x-2)$

Answer 1

$f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{f(x+h)-f\left(x\right)}{h}$

$f\left(x\right)=x^3-27$

$f\left(x\right)=x^3-27$

${lim}_{h\to 0}\frac{f(x+h)-f\left(x\right)}{h}$

$={lim}_{h\to 0}\frac{(x+h{)}^3-27-x^3+27}{h}$

$={lim}_{n\to 0}\frac{(x+h{)}^3-x^3}{h}$

$={lim}_{h\to 0}\frac{x^3+n^3+3x^{2}h+3x{h}^2-x^3}{h}$

$={lim}_{h\to 0}{h}^2+3x^2+3xh$

$=3x^2$

2)

$f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{(x+h-1)\cdot (x+h-2)-(x-1)(x-2)}{h}$

$={lim}_{h\to 0}\frac{x^2+2hx-3x+h^2-3h+2-(x^2-3x+2)}{h}$

$={lim}_{h\to 0}\frac{2hx+h^2-3h}{h}$

$=2x-3$