Derivative Calculator
Compute the derivative at any point with high-accuracy central differences, plus the tangent-line equation.
Category: Mathematics
About this tool
What Is a derivative? Derivative Calculator. The derivative calculator returns the instantaneous rate of change f'(x) at a chosen point using central differences, and draws the tangent line that matches the slope locally. The derivative measures how fast f(x) changes at one instant — geometrically, the slope of the tangent to the curve at that point. For f(x) = 3x² + 2x + 1 the derivative is f'(x) = 6x + 2. Numerically the slope is (f(x+h) − f(x−h)) / 2h with a tiny h. Called central differences, this formula is second-order accurate, so the result stabilises long before h looks small. f'(x) ≈ (f(x+h) − f(x−h)) ÷ (2h). How to Use Enter the function in x, such as 3x^2+2x+1. Enter the point x where you want the slope. Read f'(x), the function value f(x), and the tangent-line equation y = f'(a)(x − a) + f(a). Worked Example For f(x) = 3x² + 2x + 1 at x = 2, the tool evaluates points at 2 ± h and returns f'(2) = 14, matching the analytic derivative 6x + 2 evaluated at 2. It also prints the tangent y = 14(x − 2) + 17. Common Use Cases velocity from a position function marginal cost in economics optimization (finding minima and maxima) checking analytic derivative homework Important Notes Differentiation is numeric: compare against the analytic derivative for symbolic insight. The step h is scaled to the magnitude of x to reduce round-off. All computation happens in your browser. Rel