Question

given the function $f(x)=5sqrt{x^{5}}$, find $f(x)$. express your answer in radical form without using negative exponents, simplifying all fractions.

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=5\sqrt{x^{5}}$ as $f(x) = 5x^{\frac{5}{2}}$ using the rule $\sqrt[n]{a^{m}}=a^{\frac{m}{n}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For $f(x)=5x^{\frac{5}{2}}$, we have $a = 5$ and $n=\frac{5}{2}$. So $f^\prime(x)=5\times\frac{5}{2}x^{\frac{5}{2}-1}$.

Step3: Simplify the exponent and the coefficient

First, calculate $\frac{5}{2}-1=\frac{5 - 2}{2}=\frac{3}{2}$. Then, $5\times\frac{5}{2}x^{\frac{3}{2}}=\frac{25}{2}x^{\frac{3}{2}}$.

Step4: Rewrite in radical form

Using the rule $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$ again, $\frac{25}{2}x^{\frac{3}{2}}=\frac{25}{2}\sqrt{x^{3}}$.

Answer:

$\frac{25}{2}\sqrt{x^{3}}$