Question

O Valor de f'(2) para f(x)=(x^2-3x).1/3 é:
Quest.: 5
-14/3
-5/3
14/3
1/3
5/3

Answer 1

$\fbox{ f(x)=\dfrac{x^2-3x}{3} }\\\\\\\\\fbox{ f'(p)=\underset{x \to p}{\ell im}~~\dfrac{f(x)-f(p)}{x-p} }\\\\\\\\f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{\dfrac{\begin{pmatrix}x^2-3x\end{pmatrix}}{3}-\dfrac{\begin{pmatrix}2^2-3\cdot 2\end{pmatrix}}{3}}{x-2}\\\\\\\\f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{\dfrac{\begin{pmatrix}x^2-3x\end{pmatrix}}{3}-\begin{pmatrix}-\dfrac{2}{3}\end{pmatrix}}{x-2}$


$f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{\dfrac{x^2-3x}{3}+\dfrac{2}{3}}{x-2}$


$f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{\dfrac{x^2-3x+2}{3}}{x-2}\\\\\\\\(multiplicando~e~dividindo~a~express\~ao~por~3...)\\\\\\\\f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{3\cdot\hspace{-13}\diagup~~\dfrac{x^2-3x+2}{3\hspace{-6}\diagup}}{3\cdot(x-2)}\\\\\\\\f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{x^2-3x+2}{3\cdot(x-2)}}~~~~(fatorando~o~numerador...)$


$f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{(x-1)\cdot(x\hspace{-7}\diagup-2)\hspace{-11}\diagup}{3\cdot(x\hspace{-7}\diagup-2\hspace{-7}\diagup)}}\\\\\\\\f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{x-1}{3}\\\\\\\\f'(2)=\underset{x \to 2}{\ell im}~~\dfrac{2-1}{3}~~\Rightarrow~~ \fbox{ \dfrac{1}{3} }~~\checkmark$