Question

find the derivative of f(x).
$f(x) = \frac{x^7}{\sqrt6{x}}$
write your answer as a constant times a power of x.
$f(x) = \square$

Explanation:

Step1: Simplify the function

Rewrite \(\sqrt[6]{x}\) as \(x^{\frac{1}{6}}\). Then \(f(x)=\frac{x^{7}}{x^{\frac{1}{6}}}\). Using the exponent rule \(\frac{a^{m}}{a^{n}} = a^{m - n}\), we get \(f(x)=x^{7-\frac{1}{6}}=x^{\frac{42 - 1}{6}}=x^{\frac{41}{6}}\).

Step2: Differentiate using power rule

The power rule states that if \(y = x^{n}\), then \(y'=nx^{n - 1}\). For \(f(x)=x^{\frac{41}{6}}\), the derivative \(f'(x)=\frac{41}{6}x^{\frac{41}{6}-1}=\frac{41}{6}x^{\frac{41 - 6}{6}}=\frac{41}{6}x^{\frac{35}{6}}\).

Answer:

\(\frac{41}{6}x^{\frac{35}{6}}\)