Question

Encontre a derivada utilizando a definição de derivada
f'(x) = lim f(x+h)-f(x) / h h->o
a) 2x^2 - 3x
b) 1/x+1

Answer 1

a) 

$\lim_{h \to 0} \frac{2(x+h)^2-3(x+h)-2x^2+3x}{h} \\\\ \lim_{h \to 0} \frac{2(x^2+2xh+h^2)-3x-3h-2x^2+3x}{h}\\\\ \lim_{h \to 0} \frac{2x^2+4xh+2h^2-3x-3h-2x^2+3x}{h}\\\\ \lim_{h \to 0} \frac{4xh+2h^2-3h}{h}\\\\ \lim_{h \to 0} \frac{h(4x+2h-3)}{h}\\\\ \lim_{h \to 0} \ 4x+2h-3=\boxed{4x-3}$

b)

$\lim_{h \to 0} \frac{\frac{1}{x+h}+1 - \frac{1}{x}-1 }{h} \\\\ \lim_{h \to 0} \frac{\frac{x-(x+h)}{(x+h)*x}}{h} \\\\ \lim_{h \to 0} \frac{x-(x+h)}{(x+h)*x*h}}\\\\ \lim_{h \to 0} \frac{-h}{(x+h)*x*h}}\\\\ \lim_{h \to 0} \frac{-1}{(x+h)*x}}=\boxed{ \frac{-1}{x^2}}$