Question

Calcular a derivada da função: f(x) =
(4x² + 7x)5
(x2 - 9)2​

Answer 1

Definição de derivada de uma função num ponto onde x = x₀:

f'(x_{0})=\lim\limits_{h\rightarrow0}\dfrac{f(x_{0}+h)-f(x_{0})}{h}

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Primeiro, vamos simplificar a expressão, achando f(1 + h) e f(1):

f(x)=4x^{2}+5x-3\\\\f(1+h)=4(1+h)^{2}+5(1+h)-3\\f(1+h)=4(1^{2}+2\cdot1\cdot h+h^{2})+5+5h-3\\f(1+h)=4(1+2h+h^{2})+2+5h\\f(1+h)=4+8h+4h^{2}+2+5h\\f(1+h)=4h^{2}+13h+6\\\\f(1)=4(1)^{2}+5(1)-3\\f(1)=4+5-3\\f(1)=6

Então:

f'(1)=\lim\limits_{h\rightarrow0}\dfrac{f(1+h)-f(1)}{h}\\\\\\f'(1)=\lim\limits_{h\rightarrow0}\dfrac{4h^{2}+13h+6-6}{h}\\\\\\f'(1)=\lim\limits_{h\rightarrow0}\dfrac{4h^{2}+13h}{h}\\\\\\f'(1)=\lim\limits_{h\rightarrow0}\dfrac{h(4h+13)}{h}\\\\\\f'(1)=\lim\limits_{h\rightarrow0}(4h+13)\\\\\\f'(1)=4\cdot0+13\\\\\\\boxed{\boxed{f'(1)=13}}