Question

Calcule a derivada de f(x)= x^2+1 usando a definicao: f'(x)= lim f(x+h) - f(x)/h , com h tendendo a zero.

Answer 1

$\underset{h\to0}{lim}~\frac{\left((x+h)^2+1\right)-(x^2+1)}{h}~=~\underset{h\to0}{lim}~\frac{\left(x^2+h^2+2xh+1\right)-(x^2+1)}{h}~=~\\\\\\=~\underset{h\to0}{lim}~\frac{x^2+h^2+2xh+1-x^2-1}{h}~=~\underset{h\to0}{lim}~\frac{h^2+2xh}{h}~=~\underset{h\to0}{lim}~\frac{h.(h+2x)}{h}~=\\\\\\=~\underset{h\to0}{lim}~\frac{1.(h+2x)}{1}~=~\underset{h\to0}{lim}~h+2x~=~0+2x~=~\boxed{2x}$