Question

find the derivative of f(x).

f(x) = \sqrt9{x^5}

write your answer as a constant times a power of x.

f(x) = \square

Explanation:

Step1: Rewrite the function

Rewrite the radical form as an exponential form. The ninth root of \(x^5\) can be written as \(x^{\frac{5}{9}}\), so \(f(x)=x^{\frac{5}{9}}\).

Step2: Apply the power rule

The power rule for differentiation states that if \(f(x) = x^n\), then \(f^\prime(x)=nx^{n - 1}\). Here, \(n=\frac{5}{9}\), so we have \(f^\prime(x)=\frac{5}{9}x^{\frac{5}{9}-1}\).

Step3: Simplify the exponent

Calculate \(\frac{5}{9}-1=\frac{5}{9}-\frac{9}{9}=-\frac{4}{9}\). So \(f^\prime(x)=\frac{5}{9}x^{-\frac{4}{9}}\).

Answer:

\(\frac{5}{9}x^{-\frac{4}{9}}\)