Question

A derivada da função y= In 9x^3+x) em relação a variavel x no ponto x=1 é:

Answer 1

Utilizamos a regra da cadeia para achar a derivada de funções compostas:

$\boxed{\boxed{\dfrac{d}{dx}f(g(x))=f'(g(x)\cdot g'(x)}}$
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$y=ln~(9x^{3}+x)$

Se h(x) = ln x e g(x) = 9x³ + x, temos que y = h(g(x))

Derivando y:

$\dfrac{d}{dx}y=\dfrac{1}{9x^{3}+x}\cdot\dfrac{d}{dx}(9x^{3}+x)\\\\\\y'=\dfrac{1}{9x^{3}+x}\cdot(3\cdot9x^{3-1}+1\cdot x^{1-1})\\\\\\y'=\dfrac{1}{9x^{3}+x}\cdot(27x^{2}+1)\\\\\\\boxed{\boxed{y'=\dfrac{27x^{2}+1}{9x^{3}+x}}}$

Achando a derivada no ponto onde x = 1:

$y'=\dfrac{27\cdot1^{2}+1}{9\cdot1^{3}+1}=\dfrac{27+1}{9+1}=\dfrac{28}{10}\\\\\\\boxed{\boxed{y'=\dfrac{14}{5}}}$