Derivative calculator

Graph

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Formula

How it works

The derivative measures the slope of the function's graph at a point. The exact value here comes from the standard differentiation rule for the chosen family (for example, the derivative of ax² + bx + c is 2ax + b). The numerical estimate checks it by measuring the slope between two points very close to x.

Worked example

The same calculation with the default values:

• Function family Quadratic: f(x) = ax² + bx + c
• Coefficient a 1
• Coefficient b 0
• Constant c 0
• x 2

1. exact: f′(2) = 4
2. numeric estimate: f′(2) ≈ (f(x+h) − f(x−h)) ÷ 2h = 4

Result: Exact derivative f′(x): 4 · Numerical estimate: 4

Common mistakes

• Expecting an exact symbolic formula (like "2x" for x²) instead of a numerical value at the specific point entered - this calculator evaluates the slope at one point, it doesn't produce the general derivative function.
• Using too large a step size when checking this by hand - a numerical derivative's accuracy depends on using a small h; a large h gives a poor approximation of the true instantaneous rate of change.
• Evaluating the derivative at a point where the function itself is undefined or discontinuous (e.g. at x=0 for 1/x) - the numerical method can produce a misleading finite-looking number even where no true derivative exists.