Derivative Calculator
Differentiate a polynomial term by term using the power rule.
Power rule applied term by term.
The derivative gives a curve’s slope at each point. By the power rule, d/dx[a·xⁿ] = a·n·xⁿ⁻¹ — multiply by the exponent, then lower it by one. Differentiating f(x) = 3x² − 5x + 2 term by term gives f′(x) = 6x − 5.
The power rule
The derivative measures the slope of a curve — how fast f(x) is changing at each point. For a polynomial, every term follows the same simple recipe: bring the exponent down as a multiplier, then drop the exponent by one. A sum is differentiated term by term, so a whole polynomial is just a string of power-rule applications.
multiply by the exponent, then lower the exponent by one
Worked example
Differentiate f(x) = 3x² − 5x + 2:
- 1 Apply the power rule to 3x². Bring down the 2: 3 × 2 × x²⁻¹ = 6x.
- 2 Differentiate −5x. Here n = 1, so −5 × 1 × x⁰ = −5 (since x⁰ = 1).
- 3 Differentiate the constant. The derivative of 2 is 0. Combine: f′(x) = 6x − 5.
Common derivative rules
Beyond polynomials — the rules you meet next in a calculus course.
| Function f(x) | Derivative f′(x) |
|---|---|
| xⁿ | n·xⁿ⁻¹ |
| constant c | 0 |
| eˣ | eˣ |
| ln x | 1 ÷ x |
| sin x | cos x |
| cos x | −sin x |
Reading and extending the result
The derivative of a constant is zero. A flat term has no slope, so the “+ 2” vanishes — which is also why every antiderivative carries a “+ C”.
Linear terms lose their x. Because x¹ becomes x⁰ = 1, a term like −5x differentiates to the plain constant −5.
This tool handles polynomials. For trig, exponential, and logarithmic functions, use the rules in the table above — and the chain, product, and quotient rules for combinations.