Question

Find the derivative of the following functions from first principle
$(x-1)(x-2)$

Answer 1

Let $f\left(x\right)=(x-1)(x-2)$
Thus according to first principle,
$f^{{}^{\prime }}\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
$=\underset{h\to 0}{lim}\frac{(x+h-1)(x+h-2)-(x-1)(x-2)}{h}$
$=\underset{h\to 0}{lim}\frac{(x^2+hx-2x+hx+h^2-2h-x-h+2)-(x^2-2x-x+2)}{h}$
$=\underset{h\to 0}{lim}\frac{(hx+hx+h^2-2h-h)}{h}$
$=\underset{h\to 0}{lim}\frac{2hx+h^2-3h}{h}$
$=\underset{h\to 0}{lim}(2x+h-3)=(2x+0-3)=2x-3$