Question

Bom dia!
Qual a Derivada de ( a+x) × raiz de a-x ?

Answer 1

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Calcular a derivada da função

$\mathsf{f(x)=(a+x)\sqrt{a-x}}$


sendo $\mathsf{a}$ uma constante e $\mathsf{x\le a.}$

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Derivando, temos

$\mathsf{\dfrac{d}{dx}\big[f(x)\big]=\dfrac{d}{dx}\big[(a+x)\sqrt{a-x}\big]}\quad\longleftarrow\quad\textsf{(usando a Regra do Produto)}\\\\\\ =\mathsf{\dfrac{d}{dx}(a+x)\cdot \sqrt{a-x}+(a+x)\cdot \dfrac{d}{dx}\big(\!\sqrt{a-x}\big)}\\\\\\ =\mathsf{1\cdot \sqrt{a-x}+(a+x)\cdot \dfrac{d}{dx}\big[(a-x)^{1/2}\big]}\quad\longleftarrow\quad\textsf{(usando a Regra da Cadeia)}\\\\\\ =\mathsf{\sqrt{a-x}+(a+x)\cdot \dfrac{1}{2}(a-x)^{\frac{1}{2}-1}\cdot \dfrac{d}{dx}(a-x)}$

$=\mathsf{\sqrt{a-x}+(a+x)\cdot \dfrac{1}{2}(a-x)^{-1/2}\cdot (-1)}\\\\\\ =\mathsf{\sqrt{a-x}-(a+x)\cdot \dfrac{1}{2(a-x)^{1/2}}}\\\\\\ =\mathsf{\sqrt{a-x}-\dfrac{a+x}{2{\sqrt{a-x}}}}\\\\\\\\ \therefore~~\boxed{\begin{array}{c}\mathsf{\dfrac{d}{dx}\big[f(x)\big]=\sqrt{a-x}-\dfrac{a+x}{2{\sqrt{a-x}}}} \end{array}}\qquad\quad\checkmark$


Bons estudos! :-)


Tags:   derivada função real raiz quadrada polinomial regra do produto leibniz regra da cadeia cálculo diferencial