Question

calcule por definição a derivada f(x)=x²

Answer 1

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

Answer 2

Olá, Bianca!

f'(x)=$\lim_{h \to \ 0} \frac{f(x+h)-f(x)}{h} = \lim_{x\to \ 0} \frac{(x+h)^{2}-x^{2}}{h} =\\ = \lim_{h \to \ 0} \frac{x^{2}+2xh+h^{2}-x^{2}}{h} = \lim_{0 \to \ 0} \frac{2xh+h^{2}}{h} =\\ = \lim_{h \to \ 0} \frac{h(2x+h)}{h} = \lim_{h \to \ 0} (2x+h)=\\ =2x$

:)