Question

Find g′(4) given that f(4)=5, f′(4)=−1, and g(x)=(√x)*f(x).

Answer 1

Given that:

$g(x)=\sqrt[]{x}f(x)$

You need to find:

$g^{\prime}(x)$

In order to derivate the function, you need to apply the Product Rule

$\frac{d}{dx}(u\cdot v)=u\cdot v^{\prime}+v\cdot u^{\prime}$

Then, you get:

$g^{\prime}(x)=\sqrt[]{x}\cdot f^{\prime}(x)+f(x)(\sqrt[]{x})^{\prime}$

Since:

$\sqrt[]{x}=x^{\frac{1}{2}}$

You know that:

$\frac{d}{dx}(\sqrt[]{x})=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt[]{x}}$

Hence:

$\begin{gathered} g^{\prime}(x)=\sqrt[]{x}\cdot f^{\prime}(x)+f(x)(\frac{1}{2\sqrt[]{x}}) \\ \\ g^{\prime}(x)=\sqrt[]{x}\cdot f^{\prime}(x)+\frac{1}{2\sqrt[]{x}}f(x) \end{gathered}$

Knowing that you need to find:

$g^{\prime}(4)$

You can rewrite the function as follows:

$g^{\prime}(4)=\sqrt[]{4}\cdot f^{\prime}(4)+\frac{1}{2\sqrt[]{4}}f(4)$

Knowing that:

$\begin{gathered} f\mleft(4\mright)=5 \\ f^{\prime}\mleft(4\mright)=-1 \end{gathered}$

You can substitute values:

$g^{\prime}(4)=(\sqrt[]{4})(-1)+(\frac{1}{2\sqrt[]{4}})(5)$

Evaluating, you get:

$\begin{gathered} g^{\prime}(4)=(2)(-1)+(\frac{1}{2\cdot2})(5) \\ \\ g^{\prime}(4)=-\frac{3}{4} \end{gathered}$

Hence, the answer is:

$g^{\prime}(4)=-\frac{3}{4}$