Derivative Calculator
The derivative of a function at a point measures its instantaneous rate of change — geometrically, the slope of the tangent line to the graph at that point. Enter any function below and this tool estimates f′(a) numerically.
Differentiate
How the calculation works
This tool uses the central difference formula: f′(a) ≈ (f(a + h) − f(a − h)) / 2h, with a small step h. It is the same numerical method the graphing calculator uses for its tangent-line analysis, so results here match what you see when you inspect a tangent on the graph.
Numerical differentiation is an approximation. For smooth functions like polynomials, trigonometric functions, and exponentials, the estimate is accurate to many decimal places. At sharp corners (such as |x| at x = 0) or discontinuities, the derivative may not exist, and the tool will tell you so honestly instead of returning a misleading number.
What the derivative tells you
A positive derivative means the function is increasing at that point; a negative derivative means it is decreasing. The larger the magnitude, the steeper the graph. Where the derivative is zero, the graph momentarily flattens — these are the candidate locations for local maxima and minima.
Derivatives also carry physical meaning: if f(x) is position over time, f′(x) is velocity; if f(x) is velocity, f′(x) is acceleration. Try f(x) = x² at a = 2 (result: 4) and at a = −2 (result: −4) to see the sign change across the minimum.
Frequently asked questions
Is this an exact symbolic derivative?
No — this tool computes a numerical approximation using the central difference method. It is highly accurate for smooth functions but remains an estimate, shown rounded to 6 decimal places.
Why does it fail at some points?
Some functions are not differentiable everywhere: |x| has a corner at x = 0, and functions with jumps or vertical asymptotes have no meaningful slope there. The tool reports that it cannot estimate the derivative rather than guessing.
How is this related to the tangent line feature?
The tangent line to f at x = a has slope f′(a) — exactly what this tool computes. In the graphing calculator you can draw the tangent line on the graph and read off the same slope visually.