Find the derivative of a polynomial function using the power rule, and evaluate it at a specific point.
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The derivative calculator calculates the derivative of a polynomial function using the "power rule," the simplest and most commonly used differentiation rule: the derivative of the term axⁿ equals n×axⁿ⁻¹, meaning coefficient a is multiplied by exponent n, then the exponent is reduced by one. This rule is applied to each term in the polynomial separately, then the results of the terms are summed to form the complete derivative. Mathematically, the derivative represents the function's "instantaneous rate of change" at any point x, or visually, the slope of the line tangent to the function's curve at that point. After calculating the derivative's formula, the calculator also lets you evaluate it numerically at a specific x value, very useful for knowing the curve's exact slope at a given point, such as calculating instantaneous velocity from a position function in physics. A constant term in the original function always disappears entirely in the derivative, since a constant's rate of change is zero at every single point.
The slope calculator elsewhere on this site measures the steepness of a straight line between two points. A derivative extends the exact same idea — rise over run — to curves, answering the question: what is the slope of this curve at one single, exact instant, rather than averaged between two points?
The power rule makes this calculation mechanical for polynomial functions: the derivative of axⁿ is n·axⁿ⁻¹. The reasoning behind this specific rule comes from a limiting process — imagining two points on the curve getting closer and closer together until the distance between them shrinks toward zero, at which point the 'average slope' between them becomes the exact instantaneous slope at that single point. The power rule is simply the clean, general result of carrying out that limiting process for any power of x.
Applying the power rule term by term across a full polynomial works because differentiation is additive — the derivative of a sum of terms is just the sum of each term's individual derivative, with no interaction between separate terms required. This is why a derivative calculator can process an entire polynomial by handling one term at a time rather than needing to consider the whole expression together.
A constant term disappearing entirely in the derivative has an intuitive explanation: a constant, by definition, never changes as x changes, so its instantaneous rate of change is exactly zero at every point, contributing nothing to the derivative regardless of how large the constant itself is.
Physics gives the single most concrete, widely taught application of derivatives: if a function describes an object's position over time, its derivative describes the object's velocity at every instant, and the derivative of velocity in turn gives acceleration. This layered relationship — position, then velocity, then acceleration, each one the derivative of the one before it — is one of the clearest illustrations of why derivatives matter beyond pure mathematics.
Beyond physics, derivatives underpin optimization problems across economics, engineering, and business (finding a maximum or minimum value of a function by locating where its derivative equals zero), and they form the mathematical foundation of machine learning's gradient descent algorithm, which adjusts a model's parameters step by step in the direction that most rapidly reduces prediction error.
It states that the derivative of ax^n is n·ax^(n-1) — you multiply the coefficient by the exponent, then reduce the exponent by one.
No, this tool is limited to polynomial functions (terms with x raised to whole-number powers); trigonometric and logarithmic derivatives need different rules.
It represents the instantaneous rate of change of the function — for example, if f(x) describes position, f'(x) describes velocity at that instant.