Question

Resolva a derivada pela definição (1/2)x²

Answer 1

Olá

Derivada por definição

$\displaystyle \lim_{h \to 0} ~ \frac{f(x+h)-f(x)}{h}$

$f(x)= \frac{1}{2} x^2$

$\displaystyle \lim_{h \to 0} ~ \frac{ \frac{1}{2} (x+h)^2- \frac{1}{2}x^2 }{h} \\ \\ \\\text{Desenvolve o quadrado }(x+h)^2 \\ \\ \\ \lim_{h \to 0} ~ \frac{ \frac{1}{2} (x^2+2xh+h^2)- \frac{1}{2}x^2 }{h} \\ \\ \\ \text{Aplica a distributiva} \\ \\ \\ \lim_{h \to 0} ~ \frac{ \frac{1}{2} x^2+ \frac{1}{2}\cdot 2xh+ \frac{1}{2} h^2- \frac{1}{2}x^2 }{h} \\ \\ \\ \text{Simplifica}$
$\displaystyle \lim_{h \to 0} ~ \frac{\diagup\!\!\!\!\!\!\! \frac{1}{2} x^2+ \frac{1}{\diagup\!\!\!\!2}\cdot \diagup\!\!\!\!2xh+ \frac{1}{2} h^2-\diagup\!\!\!\!\!\! \frac{1}{2}x^2 }{h} \\ \\ \\ \lim_{h \to 0} ~ \frac{xh+ \frac{1}{2} h^2}{h} \\ \\ \\ \text{Poe o termo em comum em evidencia (h)} \\ \\ \\ \lim_{h \to 0} ~ \frac{h( x+ \frac{1}{2} h)}{h} \\ \\ \\ \text{Simplifica} \\ \\ \\ \lim_{h \to 0} ~ \frac{\diagup\!\!\!\!h( x+ \frac{1}{2} h)}{\diagup\!\!\!\!h}$
$\displaystyle \lim_{h \to 0} ~\left.\left(x+\dfrac{1}{2}h\,\right)~=~\left.\left(x+\dfrac{1}{2}(0)\,\right)~=~\left.\left(x+0\,\right)~=~\boxed{x}$