Question

(2) Use the correct formula above to find the derivative of the function

Answer 1

Using Product rule

Given

$f(x)=(x^6+14)\sqrt[]{x}$

Solution

Applying Product Rule

$\begin{gathered} =\frac{d}{dx}\mleft(x^6+14\mright)\sqrt{x}+\frac{d}{dx}\mleft(\sqrt{x}\mright)\mleft(x^6+14\mright) \\ \\ \\ \frac{d}{dx}(x^6+14)=6x^5 \\ \\ \frac{d}{dx}(\sqrt[]{x})=\frac{1}{2\sqrt{x}} \end{gathered}$$\frac{d}{dx}(x^6+14)\sqrt[]{x}+\frac{d}{dx}(\sqrt[]{x})(x^6+14)=6x^5\sqrt{x}+\frac{1}{2\sqrt{x}}\mleft(x^6+14\mright)$

Simplify

$=\frac{13x^6+14}{2\sqrt{x}}$

The final answer

$=\frac{13x^6+14}{2\sqrt{x}}$