Question

find the derivative of f(x).
$f(x) = \frac{x^{-9}}{\sqrt4{x}}$
write your answer as a constant times a power of x.
$f(x) = \square$

Explanation:

Step1: Simplify the function

First, rewrite the denominator $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$. Then, using the rule of exponents for division $a^m / a^n = a^{m - n}$, we have:
$f(x)=\frac{x^{-9}}{x^{\frac{1}{4}}}=x^{-9-\frac{1}{4}}=x^{-\frac{36}{4}-\frac{1}{4}}=x^{-\frac{37}{4}}$

Step2: Differentiate using power rule

The power rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$. Here, $n = -\frac{37}{4}$. So,
$f'(x)=-\frac{37}{4}x^{-\frac{37}{4}-1}=-\frac{37}{4}x^{-\frac{37}{4}-\frac{4}{4}}=-\frac{37}{4}x^{-\frac{41}{4}}$

Answer:

$-\frac{37}{4}x^{-\frac{41}{4}}$