Derivative Calculator
The derivative tells you the rate of change of a function at any point. For polynomials, the power rule makes it straightforward.
Last updated: August 2026 · Written and verified by Akash Pandey
Differentiate a polynomial
f'(x)
This runs the same calculation engine as the main scientific calculator, entirely in your browser. Nothing you type is sent anywhere.
What is a derivative?
The derivative of a function tells you its instantaneous rate of change at any point. Geometrically, it is the slope of the tangent line to the curve at that point.
The power rule
For any polynomial term axⁿ, the derivative is n·axⁿ⁻¹. Apply this to each term separately.
d/dx [axⁿ] = n·axⁿ⁻¹
- a is the coefficient
- n is the exponent
- The constant term (x⁰) differentiates to 0.
Worked examples
Find the derivative of f(x) = 3x² − 6x + 2
f(x) = 3x² − 6x + 2
- Apply the power rule to each term:
d/dx [3x²] = 2·3x¹ = 6xd/dx [−6x] = 1·(−6)x⁰ = −6d/dx [2] = 0(constant)- Combine:
f'(x) = 6x − 6.
= f'(x) = 6x − 6
A ball's height is h(t) = −5t² + 20t. What is its velocity at t = 1 second?
h(t) = −5t² + 20t, find h'(1)
- Given: a height function of time - velocity is its rate of change, i.e. its derivative.
- Method: differentiate term by term with the power rule, then substitute t = 1.
- Calculation:
h'(t) = 2·(−5)t + 20 = −10t + 20. Att = 1:−10(1) + 20 = 10. - Interpretation: the ball is rising at 10 units per second one second in. Where
h'(t) = 0(at t = 2) marks the peak of its flight, since the velocity switches from positive to negative there.
= h'(1) = 10
Real-world uses
- Velocity and acceleration - if a position function gives distance over time, its derivative gives velocity, and the derivative of that gives acceleration.
- Optimisation - the maximum or minimum of a function occurs where its derivative is zero (a flat tangent), which is how calculus finds the cheapest production cost or the largest enclosed area for a fixed perimeter.
- Marginal change in economics - "marginal cost" is the derivative of a cost function: the cost of producing one more unit, not the average cost per unit.
Common derivatives reference
| Function f(x) | Derivative f'(x) |
|---|---|
| c (constant) | 0 |
| xⁿ | nxⁿ⁻¹ |
| cx | c |
| x² | 2x |
| x³ | 3x² |
| 1/x | −1/x² |
| √x | 1/(2√x) |
Frequently Asked Questions
What is the power rule?
For any term axⁿ, the derivative is n·axⁿ⁻¹. Multiply the coefficient by the exponent, then reduce the exponent by one.
What is the derivative of a constant?
Zero. A constant does not change, so its rate of change is 0.
Can this calculator handle non-polynomial functions?
Currently it handles polynomials and basic power-rule derivatives. For more complex functions (trigonometric, exponential), use the main calculator.
What is the second derivative, and what does it mean physically?
The derivative of the derivative. For a position function, the first derivative is velocity and the second is acceleration - it tells you whether the rate of change is itself speeding up or slowing down, which the first derivative alone cannot show.
What does a derivative of zero mean?
A derivative of zero at a point means the function has a horizontal tangent there — it is flat. This often indicates a local maximum, minimum, or saddle point, which is exactly how optimisation problems use derivatives.
What is the chain rule?
The chain rule is used to differentiate composite functions: d/dx [f(g(x))] = f'(g(x)) × g'(x). It lets you break complex functions into inner and outer parts.
How does differentiation relate to integration?
They are inverse operations. The derivative of a function gives its rate of change; the integral gives the accumulated area. The Fundamental Theorem of Calculus links them.
Why does a horizontal tangent (derivative = 0) not always mean a maximum or minimum?
It can also be a saddle point, such as x³ at x = 0 - the tangent is flat there but the function keeps increasing on both sides. The second derivative (or checking either side) tells the difference.