Question

calcule a derivada de f ( x ) = x − x . e^x

Answer 1

Devemos calcular a derivada da função dada, mas antes, veja algumas regras da derivação usadas aqui:

• d/dx(x) = 1
• d/dx(x + y) = d/dx(x) + d/dx(y)
• d/dx(x . y) = d/dx(x) . y + d/dx(y) . x

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Assim derivando a função:

$\begin{array}{l}\sf f(x)=x-x\cdot e^x\\\\\sf f'(x)=\dfrac{d}{dx}(x-x\cdot e^x)\\\\\sf f'(x)=\dfrac{d}{dx}(x)+\dfrac{d}{dx}(-x\cdot e^x)\\\\\sf f'(x)=1+\dfrac{d}{dx}(-x\cdot e^x)\\\\\sf f'(x)=1+\dfrac{d}{dx}(-x)\cdot e^x+\dfrac{d}{dx}(e^x)\cdot (-x)\\\\\sf f'(x)=1-1\cdot e^x+(e^x\cdot In(e))\cdot(-x)\\\\\sf f'(x)=1-e^x+e^x\cdot1\cdot(-x)\\\\\!\boxed{\sf f'(x)=1-e^x-xe^x}\end{array}$

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Att. Nasgovaskov