Derivative Calculator
Enter a polynomial in x such as 3x^3 + 2x^2 − 5x + 7. The calculator differentiates it term by term with the power rule, shows every step, and optionally evaluates the derivative at a specific x value.
Use ^ for exponents (e.g. x^3). Only the variable x is supported. Polynomials only — no sin, cos, e^x or ln.
What is a derivative?
A derivative measures how fast a function changes. If you zoom in on the graph of f(x) at a single point, the curve starts to look like a straight line — the tangent line at that point. The slope of that tangent line is the derivative, written f′(x) or df/dx. Intuitively it answers the question "if x wiggles by a tiny amount, how much does f(x) wiggle in response?"
In everyday terms: if f(t) is the distance a car has travelled at time t, then f′(t) is its speed. If f(t) is speed, then f′(t) is acceleration. Derivatives turn "how much" into "how fast".
How this calculator differentiates, piece by piece
Each card explains one rule the tool applies to your polynomial — how a single power term shrinks, how a coefficient rides along, and how the terms split apart so each is handled on its own.
Power rule — the workhorse for x^n
Bring the exponent down as a coefficient and lower the exponent by 1. This one rule handles every term in a polynomial.
Linearity — coefficients & sums split cleanly
A constant coefficient stays put while its variable part is differentiated, and terms add or subtract independently. Any constant term becomes zero.
What the answer means — instantaneous slope
The derivative f′(x) is a new function whose value at a point equals the slope of the original curve's tangent line there — how fast f is changing at that x.
Features of this calculator
- Differentiates any polynomial in x, including negative and non-integer exponents (e.g. 4x^(-2) or 6x^(0.5))
- Applies the power rule to every term and shows the intermediate a · n · x^(n−1) form
- Combines like terms after differentiation for a clean final expression
- Optionally evaluates f′(x) at a given point to give the slope of the tangent line there
- Handles missing coefficients (x^2 as 1x^2) and constant terms correctly
- Reports the derivative of a constant as 0, so the +C in a polynomial vanishes automatically
Scope: polynomial expressions only. Trigonometric, exponential and logarithmic functions (sin x, cos x, e^x, ln x, …) are not supported — those require rules beyond the power rule.
Frequently asked questions
+Why does the constant term disappear?
Because the derivative of any constant is 0. A constant term contributes a flat horizontal piece to the graph, which has no slope, so it does not appear in f′(x). That is also why every antiderivative comes with an unknown "+C" — the constant is invisible to differentiation.
+What does evaluating the derivative at a point give me?
The slope of the tangent line to the curve at that x-value. For example, if f(x) = x^2 then f′(x) = 2x, and f′(3) = 6 — so at x = 3 the graph is climbing with slope 6.
+Can I use negative or fractional exponents?
Yes. The power rule works for any real exponent. For instance d/dx[x^(−2)] = −2·x^(−3), and d/dx[x^(1/2)] = (1/2)·x^(−1/2). Enter them as x^-2 or x^0.5.
+Why doesn't this handle sin(x) or e^x?
Those functions have their own derivative rules (d/dx[sin x] = cos x, d/dx[e^x] = e^x, and so on) that are not the power rule. To keep the steps short and the parser predictable, this calculator is intentionally scoped to polynomials only.