Question

Quanto é a derivada da função de ?


y= (x^2 +1) ^3 .ln3x ?

Answer 1

y= (x^2 +1) ^3 .ln3x
y'= 3(x^2 +1)^2 * 2x * ln3x + (x^2 +1) ^3 * 1/x

agora é fazer as manipulações

Answer 2

$\large\begin{array}{l} \textsf{Encontrar a derivada da fun\c{c}\~ao}\\\\ \mathsf{y=(x^2+1)^3\cdot \ell n(3x)}\\\\\\ \textsf{Aqui temos o produto de duas compostas:}\\\\ \left\{\!\!\begin{array}{c} \mathsf{y=f\cdot g}\\\\ \begin{array}{ll} \begin{array}{l}\mathsf{f=u^3}\\\mathsf{u=x^2+1} \end{array}\quad&\quad\begin{array}{l}\mathsf{g=\ell n(v)}\\ \mathsf{v=3x} \end{array} \end{array} \end{array}\right. \end{array}$


$\large\begin{array}{l} \textsf{Ent\~ao, combinamos a Regra do Produto com Regra}\\\textsf{da Cadeia:}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{d}{dx}(f\cdot g)}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{df}{dx}\cdot g+f\cdot \dfrac{dg}{dx}}\\\\ \mathsf{\dfrac{dy}{dx}=\left(\dfrac{df}{du}\cdot \dfrac{du}{dx}\right)\cdot g+f\cdot \left(\dfrac{dg}{dv}\cdot \dfrac{dv}{dx}\right)} \end{array}$

$\large\begin{array}{l} \mathsf{\dfrac{dy}{dx}=\left(\dfrac{d}{du}(u^3)\cdot \dfrac{d}{dx}(x^2+1)\right)\cdot \ell n(v)+u^3\cdot \left(\dfrac{d}{dv}\big(\ell n(v)\big)\cdot \dfrac{d}{dx}(3x)\right)}\\\\ \mathsf{\dfrac{dy}{dx}=\left(3u^2\cdot 2x\right)\cdot \ell n(v)+u^3\cdot \left(\dfrac{1}{v}\cdot 3\right)} \end{array}$


$\large\begin{array}{l} \textsf{Agora, substituimos de volta para a vari\'avel x:}\\\\ \mathsf{\dfrac{dy}{dx}=\big[3(x^2+1)^2\cdot 2x\big]\cdot \ell n(3x)+(x^2+1)^3\cdot \left(\dfrac{1}{3x}\cdot 3\right)}\\\\\\ \boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=6x(x^2+1)^2\cdot \ell n(3x)+(x^2+1)^3\cdot \dfrac{1}{x}} \end{array}}~~\longleftarrow~~\textsf{esta \'e a resposta.} \end{array}$


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$\large\begin{array}{l} \textsf{D\'uvidas? Comente.}\\\\\\ \textsf{Bons estudos! :-)} \end{array}$


Tags: derivada função composta produto regra da cadeia leibniz polinomial potência logarítmica log ln cálculo diferencial