Question

given the function f(x)= -\frac{3}{x}-3\sqrt{x}, find f(4). express your answer as a single fraction in simplest form.

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=-\frac{3}{x}-3\sqrt{x}$ as $f(x)= - 3x^{-1}-3x^{\frac{1}{2}}$.

Step2: Find the derivative

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $f'(x)=(-3)\times(-1)x^{-2}-3\times\frac{1}{2}x^{-\frac{1}{2}}=\frac{3}{x^{2}}-\frac{3}{2\sqrt{x}}$.

Step3: Evaluate $f'(4)$

Substitute $x = 4$ into $f'(x)$:
\[

$$\begin{align*} f'(4)&=\frac{3}{4^{2}}-\frac{3}{2\sqrt{4}}\\ &=\frac{3}{16}-\frac{3}{2\times2}\\ &=\frac{3}{16}-\frac{3}{4}\\ &=\frac{3 - 12}{16}\\ &=-\frac{9}{16} \end{align*}$$

\]

Answer:

$-\frac{9}{16}$