Power Rule Explained for Basic Derivatives
The power rule is the basic derivative rule for expressions like a*x^n. It is useful for first derivative practice because the steps are predictable: multiply by the power, then reduce the power by 1.
Open Power Rule Derivative Calculator
The basic power rule says that the derivative of a*x^n is a*n*x^(n-1). In plain terms, multiply the coefficient by the power, then subtract 1 from the power.
Use the calculators: Try the Power Rule Derivative Calculator, or use the Scientific Calculator and Fraction Calculator for supporting arithmetic.
The power rule formula
d/dx [a*x^n] = a*n*x^(n-1)
In a*x^n, a is the coefficient and n is the power. For 3x^2, the coefficient is 3 and the power is 2.
| Original expression | Power-rule step | Derivative |
|---|---|---|
| 3x^2 | 3 x 2 x x^(2-1) | 6x |
| 5x^4 | 5 x 4 x x^(4-1) | 20x^3 |
| 2x | 2 x 1 x x^(1-1) | 2 |
| 7 | Constant term | 0 |
| -4x^3 | -4 x 3 x x^(3-1) | -12x^2 |
Worked examples
Example 1: derivative of 8x^3
a = 8 n = 3 Derivative = 8 x 3 x x^(3-1) Derivative = 24x^2
Example 2: derivative of -6x^5
a = -6 n = 5 Derivative = -6 x 5 x x^(5-1) Derivative = -30x^4
Example 3: derivative of 4x
4x = 4x^1 Derivative = 4 x 1 x x^(1-1) x^0 = 1 Derivative = 4
How to use the rule step by step
- Write the expression in
a*x^nform. - Identify the coefficient
a. - Identify the power
n. - Multiply
a x n. - Reduce the power to
n - 1. - Simplify the result.
Common mistakes
- Forgetting the coefficient: In
5x^4, multiply both 5 and 4. - Dropping negative signs:
-4x^3becomes-12x^2, not12x^2. - Treating constants like powers: A standalone number such as 7 has derivative 0.
- Forgetting that x is x^1: The derivative of
2xis 2. - Using the power rule on unsupported functions: Trig, log, exponential, implicit, and product/quotient cases need other rules.
When to use another calculator
- Use the Power Rule Derivative Calculator for one-term power-rule checks.
- Use the Scientific Calculator for arithmetic, exponents, roots, and expression checks.
- Use the Fraction Calculator when coefficients or exponents involve fractions.
Limits of the current derivative tool
Calzivo currently supports single-term power-rule expressions of the form a*x^n. It does not solve trig, logarithmic, exponential, implicit, product-rule, quotient-rule, chain-rule, or multi-term symbolic derivatives.
What is the power rule in simple words?
Multiply the coefficient by the power, then reduce the power by 1.
What is the derivative of x?x is the same as x^1, so its derivative is 1.
What is the derivative of a constant?
A constant has derivative 0 because it does not change with x.
Can I use the power rule for negative coefficients?
Yes. Keep the negative sign when multiplying by the power.
Does the power rule handle every derivative?
No. It is a beginner rule for power expressions. Other functions need other derivative rules.
For a*x^n, multiply a by n and reduce the power by 1. The current derivative tool only supports that basic pattern.
Use the tool instead
Use the matching calculator when you want to plug in your own numbers and get a result faster.
Open Calculator