Why Finite Differences?

When a function lacks an easy analytic derivative, we often turn to finite differences. The idea is simple: sample $f$ at points near $x\_0$ and use these values to estimate the slope. The forward formula $\frac{f\left(x\_0+h\right)-f\left(x,\_0\right)}{h}$ gives a first-order approximation. Backward difference uses $f\left(x,\_0\right)-f\left(x\_0-h\right)$, while the central difference $\frac{f\left(x\_0+h\right)-f\left(x\_0-h\right)}{2h}$ generally offers higher accuracy.

How to Use This Tool

Type in your function using math.js syntax along with the evaluation point $x\_0$ and a small step size $h$. Choose a method and press Approximate to obtain the derivative. The smaller $h$ is, the closer the approximation is to the true derivative, but very tiny values may introduce rounding errors due to floating-point limitations.

Applications

Finite difference methods underpin many numerical algorithms, from solving differential equations to simulating physical systems. Engineers and scientists often discretize continuous problems using these techniques when closed-form solutions are unavailable.