The function you want to differentiate is the cube root of \(x\), which can be expressed as:

\[
f(x) = \sqrt[3]{x} = x^{1/3}
\]

To find the derivative of this function, we use the power rule for derivatives. The power rule states that for any function \(x^n\), its derivative is given by:

\[
\frac{d}{dx}(x^n) = n x^{n-1}
\]

Applying this rule to \(f(x) = x^{1/3}\):

\[
f'(x) = \frac{1}{3} x^{1/3 - 1} = \frac{1}{3} x^{-2/3}
\]

Thus, the derivative of \(f(x) = \sqrt[3]{x}\) is:

\[
f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \sqrt[3]{x^2}}
\]

Would you like more details or have any other questions?

Here are 5 related questions you can explore:
1. What is the second derivative of \(f(x) = \sqrt[3]{x}\)?
2. How do you compute the derivative of \(f(x) = \sqrt[3]{x^2}\)?
3. How do you differentiate \(f(x) = x^{m/n}\) for any general rational power?
4. Can you apply the chain rule to differentiate \(\sqrt[3]{g(x)}\)?
5. How do you compute the derivative of implicit functions involving cube roots?

**Tip:** When dealing with fractional powers, always check if applying logarithmic differentiation simplifies the process.

Discover how to differentiate the cube root of x using the power rule. This detailed explanation breaks down the steps for applying the power rule to the function f(x) = x^{1/3} and provides additional related questions for deeper understanding. Ideal for students in Grades 11-12 or introductory college calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation