Question

question given the function f(x)= -\frac{3\sqrt5{x^{7}}}{4}, 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 $f(x)=-\frac{3\sqrt[4]{x^{5}}}{x^{7}}$ using exponent - rules. $\sqrt[4]{x^{5}}=x^{\frac{5}{4}}$, so $f(x)=-3\frac{x^{\frac{5}{4}}}{x^{7}}$. Then, by the rule $\frac{x^{m}}{x^{n}}=x^{m - n}$, we have $f(x)=-3x^{\frac{5}{4}-7}=-3x^{\frac{5 - 28}{4}}=-3x^{-\frac{23}{4}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For $f(x)=-3x^{-\frac{23}{4}}$, we have $a=-3$ and $n =-\frac{23}{4}$. So $f^\prime(x)=-3\times(-\frac{23}{4})x^{-\frac{23}{4}-1}$.

Step3: Simplify the exponent and coefficient

First, calculate the coefficient: $-3\times(-\frac{23}{4})=\frac{69}{4}$. Then, calculate the exponent: $-\frac{23}{4}-1=-\frac{23 + 4}{4}=-\frac{27}{4}$. So $f^\prime(x)=\frac{69}{4}x^{-\frac{27}{4}}$.

Step4: Rewrite without negative exponents

Using the rule $x^{-n}=\frac{1}{x^{n}}$, we rewrite $f^\prime(x)$ as $f^\prime(x)=\frac{69}{4x^{\frac{27}{4}}}$. And since $x^{\frac{27}{4}}=\sqrt[4]{x^{27}}$, we have $f^\prime(x)=\frac{69}{4\sqrt[4]{x^{27}}}$.

Answer:

$\frac{69}{4\sqrt[4]{x^{27}}}$