1. What is the Power Rule?

The Power Rule is a basic differentiation rule in calculus that allows us to find the derivative of a function of the form \( f(x) = x^n \) where \( n \) is any real number exponent.

2. How Does the Power Rule Work?

The Power Rule formula is:

Where:

• \( n \) — The exponent (can be any real number)
• \( x \) — The variable of differentiation

Explanation: To apply the power rule, you bring down the exponent as a coefficient and then subtract one from the original exponent.

3. Importance of the Power Rule

Details: The Power Rule is fundamental in calculus as it provides a quick method to differentiate polynomial functions and is the basis for more complex differentiation rules.

4. Using the Calculator

Tips: Enter the exponent (n) and the variable (default is x). The calculator will show the derivative using the power rule.

5. Frequently Asked Questions (FAQ)

Q1: Does the power rule work for all exponents?
A: Yes, the power rule works for all real number exponents, including fractions and negative numbers.

Q2: How is this different from the chain rule?
A: The power rule is for simple \( x^n \) terms, while the chain rule is used for composite functions like \( (3x+1)^n \).

Q3: What's the derivative of a constant?
A: The derivative of any constant (which can be seen as \( x^0 \)) is 0.

Q4: Can this be used for integration?
A: There's a similar but reversed power rule for integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) (for \( n \neq -1 \)).

Q5: What about exponents that are variables?
A: For functions like \( x^x \), you need logarithmic differentiation, not the simple power rule.