Question

given the function f(x) = 6√x, find f(x). express your answer in radical form without using negative exponents, simplifying all fractions. answer attempt 1 out of 2 f(x) =

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=6\sqrt{x}$ as $f(x) = 6x^{\frac{1}{2}}$ using the rule $\sqrt{x}=x^{\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y'=anx^{n - 1}$. For $f(x)=6x^{\frac{1}{2}}$, we have $a = 6$ and $n=\frac{1}{2}$. So $f'(x)=6\times\frac{1}{2}x^{\frac{1}{2}-1}$.

Step3: Simplify the exponent and coefficient

$6\times\frac{1}{2}=3$, and $\frac{1}{2}-1=\frac{1 - 2}{2}=-\frac{1}{2}$. So $f'(x)=3x^{-\frac{1}{2}}$.

Step4: Rewrite in radical form

Using the rule $x^{-\frac{1}{2}}=\frac{1}{\sqrt{x}}$, we get $f'(x)=\frac{3}{\sqrt{x}}$.

Answer:

$\frac{3}{\sqrt{x}}$