How to Use This Calculator
Find the derivative of f(x) = x³ at x = 2. Enter "x^3" as the function, set Point x = 2, keep Step Size h = 0.001, and click "Central Difference". The result shows f'(2) ≈ 12.000001. The exact derivative of x³ is 3x², so f'(2) = 3 × 4 = 12. The numerical result is nearly perfect — the tiny error (0.000001) comes from the finite step size. The step-by-step output shows: f(2) = 8, f(2.001) = 8.012006, f(1.999) = 7.988006, then the formula (8.012006 - 7.988006) / (2 × 0.001) = 12.000001. Now try "Forward Difference" — you get about 12.006, which is less accurate for the same h because forward difference has larger error (O(h) vs O(h²)).
About Derivative Calculator
This derivative calculator computes the numerical derivative of any function at a specified point using two common finite difference methods. The central difference method (f(x+h) - f(x-h)) / (2h) offers second-order accuracy (error O(h)), while the forward difference method (f(x+h) - f(x)) / h provides first-order accuracy (error O(h)). Choose a smaller step size h for more precise results, though extremely small values may introduce floating-point rounding errors. This tool is invaluable for calculus students learning differentiation, engineers performing sensitivity analysis, and scientists calculating rates of change in their data. The step-by-step breakdown shows each evaluation point and the formula application, making it an excellent learning aid for understanding how numerical derivatives work.
When to Use This Calculator
Physics (Instantaneous Velocity): If position s(t) = 4.9t² meters (free fall), the velocity is the derivative s'(t) = 9.8t. Enter "4.9*t^2", set x = 5 (seconds), and the derivative at t=5 should be 49 m/s. This is exactly how physicists calculate instantaneous velocity from position data.
Economics (Marginal Analysis): If a company's profit function is P(x) = -0.5x² + 100x - 500 (where x is units produced), the marginal profit at x = 60 is the derivative. Enter "-0.5*x^2 + 100*x - 500", x = 60. The derivative tells you the profit increase from producing one more unit. In this case, marginal profit at x=60 is 40.
Engineering (Sensitivity Analysis): When a bridge design's stress σ depends on load L as σ(L) = 0.02L² + 0.5L, the rate of stress change at L = 1000 kN tells engineers the safety margin. Enter "0.02*x^2 + 0.5*x", x = 1000, and the derivative shows how quickly stress increases with additional load.