The Power Rule for Finding the Derivative of a Cube Root

In calculus, the derivative of a function represents its instantaneous rate of change. When dealing with functions involving cube roots, we can leverage the power rule to find their derivatives effectively. This article will guide you through the process, explaining the concepts and providing illustrative examples.

Understanding the Power Rule

The power rule is a fundamental principle in calculus that states:

**d/dx (xⁿ) = n*x^(n-1)**

Where:

• d/dx represents the derivative with respect to x.
• x is the variable.
• n is any real number.

Applying the Power Rule to Cube Roots

Cube roots can be expressed as fractional exponents. Remember that the cube root of a number is the same as raising that number to the power of 1/3. For instance:

∛x = x^1/3

Therefore, to find the derivative of a cube root using the power rule, we follow these steps:

1. **Rewrite the cube root as a fractional exponent.**
2. **Apply the power rule.**
3. **Simplify the result.**

Example: Finding the Derivative of ∛x²

Let's find the derivative of the function f(x) = ∛x².

1. **Rewrite the cube root:** f(x) = x^2/3
2. **Apply the power rule:** f'(x) = (2/3)x^{(2/3 - 1)}
3. **Simplify:** f'(x) = (2/3)x^(-1/3)

Therefore, the derivative of ∛x² is (2/3)x^(-1/3).

Key Points to Remember

• Cube roots are expressed as x^1/3.
• The power rule applies to fractional exponents.
• Simplify the result after applying the power rule.

Conclusion

The power rule is a powerful tool for finding derivatives of functions involving cube roots. By understanding the concept of fractional exponents and applying the power rule correctly, you can efficiently calculate derivatives and analyze the rate of change of these functions.