The derivative of a function measures its instantaneous rate of change at any point. For polynomial functions, the power rule makes differentiation straightforward: the derivative of axⁿ is n·axⁿ⁻¹. This calculator accepts any polynomial expression in x, applies the power rule to each term, and returns the first derivative f'(x) and the second derivative f''(x). It also evaluates f'(x) at a specific x value when you want the slope at a particular point, allowing you to work out instantaneous rates without carrying out the algebra manually.

The Power Rule for Polynomials

The power rule states: d/dx [axⁿ] = n·axⁿ⁻¹. You multiply the coefficient by the exponent and then reduce the exponent by one. For polynomials, you apply this rule to each term independently using the sum rule: the derivative of a sum is the sum of the derivatives. As a result, differentiating a polynomial term by term is a direct and mechanical process. Given that constant terms have an exponent of zero, their derivative is zero (they disappear from f'(x)). The Khan Academy differentiation rules review covers the power rule and sum rule with animated examples showing each step.

The Second Derivative and Concavity

The second derivative f''(x) is the derivative of f'(x). It measures the rate of change of the slope itself, which determines the concavity of the function. Where f''(x) is positive, the graph is concave up (bowl-shaped); where f''(x) is negative, it is concave down (hill-shaped). With that in mind, the second derivative is used to classify critical points: if f'(x) = 0 and f''(x) is positive at that point, the function has a local minimum there; if f''(x) is negative, there is a local maximum. This calculator returns f''(x) automatically alongside f'(x), so you can figure out concavity without a separate differentiation step.

A point where f''(x) changes sign is called an inflection point, where the curve changes from concave up to concave down or vice versa. On top of that, the second derivative is central to physics: if position is given as a polynomial in time, the first derivative gives velocity and the second derivative gives acceleration. The Cuemath power rule guide explains how the rule extends to negative and fractional exponents, which goes beyond this calculator's polynomial scope but is useful context for further study.

Evaluating the Derivative at a Point

The slope of a polynomial at a specific x value is found by substituting x into f'(x). For f(x) = x³ - 3x, f'(x) = 3x² - 3. At x = 2, f'(2) = 3(4) - 3 = 9, meaning the function is rising at a rate of 9 units per unit of x at that point. Given that the tangent line to the curve at x = 2 has slope 9, this also lets you build up the equation of the tangent line using point-slope form. The calculator returns f'(x_value) directly when you enter an evaluation point, so you can narrow down the slope at any x without substituting manually.

That said, evaluating the derivative is only meaningful where the polynomial is defined, which for polynomials is all real x. Our Integral Calculator is the companion tool for the reverse operation: finding F(x) such that F'(x) = f(x). Our Chain Rule Calculator extends differentiation to composite polynomial functions of the form (ax^m + b)^n, which the power rule alone cannot handle. The CalcWorkshop power rule tutorial provides nine step-by-step examples building from simple to complex polynomial differentiation.

Applying Derivatives to Motion and Rates

In physics, the derivative gives velocity from a position function and acceleration from a velocity function. If a particle's position is s(t) = 4t³ - 9t² + 6t, then v(t) = s'(t) = 12t² - 18t + 6, and acceleration is a(t) = v'(t) = 24t - 18. Given that these calculations follow directly from the power rule, it is straightforward to carry out derivative analysis on any polynomial motion problem using this calculator. With that in mind, the values of t where v(t) = 0 are the moments when the particle is stationary, and they can be found by solving the derivative equation using factoring or the quadratic formula. On top of that, engineers carry out derivative calculations to figure out where stress concentrations are maximum in beams or structures, since the maximum stress corresponds to a critical point of the stress polynomial. These are real applications where the calculator's ability to return f'(x) and f''(x) together is directly useful.