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Sagot :
Given that:
[tex]g(x)=\sqrt[]{x}f(x)[/tex]You need to find:
[tex]g^{\prime}(x)[/tex]In order to derivate the function, you need to apply the Product Rule
[tex]\frac{d}{dx}(u\cdot v)=u\cdot v^{\prime}+v\cdot u^{\prime}[/tex]Then, you get:
[tex]g^{\prime}(x)=\sqrt[]{x}\cdot f^{\prime}(x)+f(x)(\sqrt[]{x})^{\prime}[/tex]Since:
[tex]\sqrt[]{x}=x^{\frac{1}{2}}[/tex]You know that:
[tex]\frac{d}{dx}(\sqrt[]{x})=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt[]{x}}[/tex]Hence:
[tex]\begin{gathered} g^{\prime}(x)=\sqrt[]{x}\cdot f^{\prime}(x)+f(x)(\frac{1}{2\sqrt[]{x}}) \\ \\ g^{\prime}(x)=\sqrt[]{x}\cdot f^{\prime}(x)+\frac{1}{2\sqrt[]{x}}f(x) \end{gathered}[/tex]Knowing that you need to find:
[tex]g^{\prime}(4)[/tex]You can rewrite the function as follows:
[tex]g^{\prime}(4)=\sqrt[]{4}\cdot f^{\prime}(4)+\frac{1}{2\sqrt[]{4}}f(4)[/tex]Knowing that:
[tex]\begin{gathered} f\mleft(4\mright)=5 \\ f^{\prime}\mleft(4\mright)=-1 \end{gathered}[/tex]You can substitute values:
[tex]g^{\prime}(4)=(\sqrt[]{4})(-1)+(\frac{1}{2\sqrt[]{4}})(5)[/tex]Evaluating, you get:
[tex]\begin{gathered} g^{\prime}(4)=(2)(-1)+(\frac{1}{2\cdot2})(5) \\ \\ g^{\prime}(4)=-\frac{3}{4} \end{gathered}[/tex]Hence, the answer is:
[tex]g^{\prime}(4)=-\frac{3}{4}[/tex]
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