Question

Determine a derivada de f'(x) no ponto Xo indicado:

f(x) = (5x² + 3x - 9)/(x² + 5) para Xo= 5

gabarito f'(5) = 4/9

Answer 1

Olá

$\displaystyle \mathsf{f(x)= \frac{5x^2+3x-9}{x^2+5} }$

Temos que utilizar a regra do produto, dada por:

$\displaystyle \mathsf{ \left(\frac{f}{g}\right)'= \frac{f'\cdot g ~-~f\cdot g'}{g^2} }$


Derivando

$\displaystyle \mathsf{f'(x)= \frac{(10x+3)\cdot (x^2+5)~-~ (5x^2+3x-9)\cdot (2x)}{(x^2+5)^2} }$

Simplificando para facilitar nas contas.

OBS: Já poderia substituir x por 5


$\displaystyle \mathsf{f'(x)= \frac{(10x^3+50x+3x^2+15)~-~(10x^3+6x^2-18x)}{(x^2+5)^2} }\\\\\\\mathsf{f'(x)= \frac{10x^3+50x+3x^2+15~-~10x^3-6x^2+18x}{(x^2+5)^2} }\\\\\\\boxed{\mathsf{ f'(x)=\frac{-3x^2+68x+15}{(x^2+5)^2} }}$


Substituindo no ponto xo

$\displaystyle \mathsf{ f'(5)=\frac{-3\cdot(5)^2+68\cdot (5)+15}{((5)^2+5)^2} }\\\\\\\mathsf{f'(5)= \frac{-3\cdot 25 +340 + 15 }{30^2} }\\\\\\\mathsf{f'(5)= \frac{400}{900} }\\\\\\\boxed{\mathsf{f'(5)= \frac{4}{9} }}$