Question

question given the function f(x) = 1 / (2√x), find f(x). express your answer in radical form without using negative exponents, simplifying all.

Explanation:

Step1: Rewrite the function

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

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $f(x)=\frac{1}{2}x^{-\frac{1}{2}}$, we have $a=\frac{1}{2}$ and $n =-\frac{1}{2}$. So $f^\prime(x)=\frac{1}{2}\times(-\frac{1}{2})x^{-\frac{1}{2}-1}=-\frac{1}{4}x^{-\frac{3}{2}}$.

Step3: Convert back to radical form

Rewrite $-\frac{1}{4}x^{-\frac{3}{2}}$ as $-\frac{1}{4x^{\frac{3}{2}}}=-\frac{1}{4\sqrt{x^{3}}}$.

Answer:

$-\frac{1}{4\sqrt{x^{3}}}$