Question

Usando a definição, calcule a derivada da função f(x) = 3x + 1.

Answer 1

$f(x) = 3x + 1$

$f'(x) = \dfrac{d}{d \: \: x}( 3x + 1)$

$f'(x) = \dfrac{d}{d \: \: x} (3x) + \dfrac{d}{d \: \: x} (1)$

$f'(x) = 3 + 0$

$\boxed{f'(x) = 3}$

Answer 2

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Definição de Derivada

$\huge\boxed{\boxed{\boxed{\boxed{\displaystyle\sf f'(x)=\lim_{h \to 0}\dfrac{f(x+h)-f(x)}{h}}}}}$

$\large\boxed{\begin{array}{l}\sf f(x)=3x+1\\\displaystyle\sf f'(x)=\lim_{h \to 0}\dfrac{3\cdot(x+h)+1-[3x+1]}{h}\\\displaystyle\sf f'(x)=\lim_{ h \to 0}\dfrac{\diagdown\!\!\!\!\!\!3x+3h+\diagup\!\!\!1-\diagdown\!\!\!\!\!\!3x-\diagup\!\!\!1}{h}\\\displaystyle\sf f'(x)=\lim_{ h \to 0}\dfrac{3\diagup\!\!\!h}{\diagup\!\!\!h}\\\displaystyle\sf f'(x)=\lim_{h \to 0}3\\\sf f'(x)=3\end{array}}$