Question

find the derivative of f(x).
$f(x) = \frac{\sqrt3{x}}{x^8}$
write your answer as a constant times a power of x.
$f(x) = \square$

Explanation:

Step1: Simplify the function

First, rewrite the cube root and the denominator using exponents. The cube root of \( x \) is \( x^{\frac{1}{3}} \), and \( \frac{1}{x^8} = x^{-8} \). So, \( f(x)=\frac{x^{\frac{1}{3}}}{x^8} \). Using the exponent rule \( \frac{a^m}{a^n}=a^{m - n} \), we get \( f(x)=x^{\frac{1}{3}-8} = x^{\frac{1}{3}-\frac{24}{3}}=x^{-\frac{23}{3}} \).

Step2: Differentiate using the power rule

The power rule for differentiation is \( \frac{d}{dx}(x^n)=nx^{n - 1} \). Here, \( n = -\frac{23}{3} \). So, \( f'(x)=-\frac{23}{3}x^{-\frac{23}{3}-1} \). Calculate the exponent: \( -\frac{23}{3}-1 = -\frac{23}{3}-\frac{3}{3}=-\frac{26}{3} \). Thus, \( f'(x)=-\frac{23}{3}x^{-\frac{26}{3}} \).

Answer:

\( -\frac{23}{3}x^{-\frac{26}{3}} \)