Question

8. Usando a definição, determinar a derivada das seguintes funções:
D)F(x)=1-x/x+3

Answer 1

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

$\boxed{\begin{array}{l}\displaystyle\sf F'(x)=\lim_{h \to 0}\dfrac{1}{h}\bigg(\dfrac{x+3-x^2-3x-hx-3h-(x+h+3-x^2-hx-3x)}{(x+3)(x+h+3)}\bigg)\\\\\displaystyle\sf F'(x)=\lim_{h \to 0}\dfrac{1}{h}\bigg(\dfrac{\diagup\!\!\!x+\diagup\!\!\!\!3-\diagup\!\!\!x^2-\diagdown\!\!\!\!\!\!3x-\diagdown\!\!\!\!\!hx-3h-\diagup\!\!\!x-h-\diagup\!\!\!3+\diagup\!\!\!\!x^2+\diagdown\!\!\!\!hx+\diagdown\!\!\!\!\!\!3x}{(x+3)(x+h+3)}\bigg)\end{array}}$

$\Large\boxed{\begin{array}{l}\displaystyle\sf F'(x)=\lim_{h \to 0}\dfrac{1}{\backslash\!\!\!h}\cdot\dfrac{-4\backslash\!\!\!h}{(x+3)(x+h+3}\\\\\displaystyle\sf F'(x)=\lim_{h \to 0}\dfrac{-4}{(x+3)(x+h+3)}\\\\\sf F'(x)=\dfrac{-4}{(x+3)(x+0+3)}\\\huge\boxed{\boxed{\boxed{\boxed{\sf F'(x)=-\dfrac{4}{(x+3)^2}}}}}\end{array}}!$