Question

Dê as derivadas das funções abaixo ?
a) y= tg( ln x)
b) y= ln( tg x)

Answer 1

$\large\begin{array}{l} \textsf{a) }\mathsf{y=tg(\ell n\,x)}\\\\ \textsf{Expressando y como uma fun\c{c}\~ao composta:}\\\\ \left\{\! \begin{array}{l} \mathsf{y=tg\,u}\\ \mathsf{u=\ell n\,x} \end{array} \right. \end{array}$


$\large\begin{array}{l} \textsf{Deriva-se usando a Regra da Cadeia:}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{d}{du}(tg\,u)\cdot \dfrac{d}{dx}(\ell n\,x)}\\\\ \mathsf{\dfrac{dy}{dx}=sec^2\,u\cdot \dfrac{1}{x}}\\\\ \mathsf{\dfrac{dy}{dx}=sec^2(\ell n\,x)\cdot \dfrac{1}{x}}\\\\ \boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=\dfrac{sec^2(\ell n\,x)}{x}} \end{array}}\quad\longleftarrow\quad\textsf{esta \'e a resposta.} \end{array}$

_________


$\large\begin{array}{l} \textsf{b) }\mathsf{y=\ell n(tg\,x)}\\\\ \textsf{Expressando y como uma fun\c{c}\~ao composta:}\\\\ \left\{\! \begin{array}{l} \mathsf{y=\ell n\,v}\\ \mathsf{v=tg\,x} \end{array} \right. \end{array}$


$\large\begin{array}{l} \textsf{Deriva-se usando a Regra da Cadeia:}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{dy}{dv}\cdot \dfrac{dv}{dx}}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{d}{dv}(\ell n\,v)\cdot \dfrac{d}{dx}(tg\,x)}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{1}{v}\cdot sec^2\,x}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{1}{tg\,x}\cdot sec^2\,x}\\\\ \mathsf{\dfrac{dy}{dx}=cotg\,x\cdot sec^2\,x}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{cos\,x}{sen\,x}\cdot \left(\dfrac{1}{cos\,x}\right)^{\!2}}\end{array}$

$\large\begin{array}{l} \mathsf{\dfrac{dy}{dx}=\dfrac{~\diagup\!\!\!\!\!\!\! cos\,x}{sen\,x}\cdot \dfrac{1}{cos^{\!\diagup\!\!\!\! 2}\,x}}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{1}{sen\,x}\cdot \dfrac{1}{cos\,x}}\\\\ \boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=cossec\,x\cdot sec\,x} \end{array}}\quad\longleftarrow\quad\textsf{esta \'e a resposta.} \end{array}$


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


Tags: derivada função composta regra da cadeia tangente tg logaritmo ln cálculo diferencial