Power Rule Derivative Calculator
Differentiate any single-term function in the form f(x) = a·xn using the calculus power rule.
What Is the Power Rule?
The power rule is one of the most useful ideas in differential calculus. It gives you a fast, reliable way to find the derivative of power functions like x2, x5, or even x-3.
Power rule formula: if f(x) = a·xn, then f’(x) = a·n·xn-1.
In words: multiply by the exponent, then reduce the exponent by 1.
How to Use This Derivative Calculator
- Enter the coefficient a (for example, 7 in 7x4).
- Enter the exponent n (for example, 4 in 7x4).
- Optionally enter a specific x-value to evaluate the derivative numerically.
- Click Calculate Derivative to get the symbolic derivative and steps.
Worked Examples
Example 1: f(x) = 3x5
Multiply coefficient and exponent: 3 × 5 = 15. Subtract 1 from exponent: 5 - 1 = 4. So the derivative is f’(x) = 15x4.
Example 2: f(x) = -2x-3
Multiply: -2 × -3 = 6. Subtract 1: -3 - 1 = -4. So f’(x) = 6x-4.
Example 3: f(x) = 9x
Here the exponent is 1. Multiply: 9 × 1 = 9. New exponent: 1 - 1 = 0. Since x0 = 1, the derivative simplifies to f’(x) = 9.
Common Mistakes to Avoid
- Forgetting to reduce the exponent by 1 after multiplying.
- Dropping negative signs when coefficients or exponents are negative.
- Thinking constants have complicated derivatives (they do not; derivative of a constant is 0).
- Confusing the original function with its derivative during simplification.
When This Power Rule Tool Is Most Useful
This calculator is ideal for students in algebra, precalculus, AP Calculus, and college calculus who want to check homework or build confidence. It is also useful for quick derivative verification when working on optimization, rate of change, and tangent line problems.
Related Calculus Concepts
The power rule often appears alongside:
- Constant multiple rule (differentiate c·f(x)).
- Sum and difference rules (differentiate term-by-term).
- Product and quotient rules for multi-term expressions.
- Chain rule for composite functions like (3x + 1)5.
Quick FAQ
Can this calculator handle decimal exponents?
Yes. You can enter fractional or decimal exponents such as 0.5 or -1.25.
What is the derivative of a constant?
If n = 0, then f(x) = a and the derivative is always 0.
Can I evaluate the derivative at a point?
Yes. Enter a value for x and the calculator will return both the symbolic derivative and the numeric derivative value.