Answer to Question #258023 in Calculus for Russell

Question

find f(x) = x² - 2x -2 find f' (x) using the long method

Answer 1

$\displaystyle f(x) = x^2 -2x -2\\ f'(x) = \lim _{h \to 0}\frac{|f(x+h)-f(x)|}{|h|} \\ \text{where $f(x+h) = (x+h)^2 -2(x+h) -2$ }\\ =x^2+2xh+h^2-2x-2h-2\\ \therefore \frac{f(x+h)-f(x)}{h}=2x+h-2\\ \therefore \lim_{h \to 0}\frac{|f(x+h)-f(x)|}{|h|}=2x-2$