Question

find the derivative of f(x).
$f(x) = \sqrt6{x^{-5}}$
write your answer as a constant times a power of x.
$f(x) = \square$

Explanation:

Step1: Rewrite the radical as a power

Recall that \(\sqrt[n]{x^m}=x^{\frac{m}{n}}\). So for \(f(x)=\sqrt[6]{x^{-5}}\), we can rewrite it as \(f(x)=x^{\frac{-5}{6}}\).

Step2: Apply the power rule for differentiation

The power rule states that if \(f(x)=x^n\), then \(f'(x)=nx^{n - 1}\). Here, \(n =-\frac{5}{6}\).
So, \(f'(x)=-\frac{5}{6}x^{-\frac{5}{6}-1}\).

Step3: Simplify the exponent

Calculate \(-\frac{5}{6}-1=-\frac{5}{6}-\frac{6}{6}=-\frac{11}{6}\).
Thus, \(f'(x)=-\frac{5}{6}x^{-\frac{11}{6}}\).

Answer:

\(-\frac{5}{6}x^{-\frac{11}{6}}\)