Question

Calcule a derivadaa da função f(x)=1/x² ; x=2

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\rm De~\!\!finic_{\!\!,}\tilde ao\,de\,derivada\,no\,ponto}\\\displaystyle\sf f'\!(p)=\lim_{x \to p}\dfrac{f(x)-f(p)}{x-p}\\\\\sf f(x)=\dfrac{1}{x^2}\\\\\sf f(2)=\dfrac{1}{2^2}=\dfrac{1}{4}\\\\\displaystyle\sf f'(2)=\dfrac{\frac{1}{x^2}-\dfrac{1}{4}}{x-2}\\\\\sf f'(2)=\dfrac{1}{(x-2)}\cdot\bigg(\dfrac{4-x^2}{4x^2}\bigg)\end{array}}$

$\Large\boxed{\begin{array}{l}\displaystyle\sf f'(2)=\lim_{x \to 2}\dfrac{1}{(x-2)}\cdot\bigg(\dfrac{(2-x)(2+x)}{4x^2}\bigg)\\\\\displaystyle\sf f'(2)=-\lim_{x \to 2}\dfrac{1}{\diagdown\!\!\!\!\!(x-\diagdown\!\!\!\!\!2)}\cdot\bigg(\dfrac{\diagdown\!\!\!\!\!(x-\diagdown\!\!\!\!\!2)\cdot(x+2)}{4x^2}\bigg)\\\\\displaystyle\sf f' (2)=-1\cdot\lim_{x \to 2}\dfrac{x+2}{4x^2}\\\\\sf f'(2)=-1\cdot\dfrac{2+2}{4\cdot2^2}\\\\\sf f'(2)=-1\cdot\dfrac{\backslash\!\!\!4}{\backslash\!\!\!4\cdot4}\\\\\Huge\boxed{\boxed{\boxed{\boxed{\sf f'(2)=-\dfrac{1}{4}}}}}\end{array}}$