1. What is the Power Rule?

The Power Rule is a basic rule in calculus for finding the derivative of a function of the form \( f(x) = x^n \), where \( n \) is any real number. It's one of the most fundamental differentiation rules.

2. How Does the Formula Work?

The Power Rule formula is:

Where:

• \( n \) — The exponent (can be any real number)
• \( x \) — The variable of differentiation
• \( \frac{d}{dx} \) — The derivative operator

Explanation: To find the derivative, multiply the original exponent by the coefficient and subtract one from the exponent.

3. Importance of the Power Rule

Details: The Power Rule is essential for basic differentiation and serves as the foundation for more complex differentiation rules. It's used extensively in physics, engineering, economics, and other fields involving rates of change.

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 formula.

5. Frequently Asked Questions (FAQ)

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

Q2: What if the exponent is zero?
A: If n=0, the derivative is 0 (since the derivative of any constant is 0).

Q3: Can the power rule be used with other differentiation rules?
A: Yes, it's often combined with the product rule, quotient rule, and chain rule for more complex functions.

Q4: How is the power rule derived?
A: It can be derived using the limit definition of the derivative and the binomial theorem.

Q5: Does this work for variables other than x?
A: Yes, the power rule works for any variable (t, y, etc.) as long as you're differentiating with respect to that variable.