Higher Derivative Calculator

Calculator Input

Function f(x)

Use ^ for powers. Example: sin(x)*exp(x), x^4+2*x^3-x, log(x), sqrt(x+4).

Variable

Single variable recommended, such as x or t.

Derivative Order

Allowed range: 1 to 12.

Evaluation Point

The derivative is evaluated at this point.

Step Size h

Smaller values can improve or worsen stability.

Stencil Radius

Total stencil points = 2r + 1.

Graph Minimum x

Graph Maximum x

Graph Points

More points give smoother plots and slower runs.

Display Precision

Supported functions: sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, log, ln, log10, sqrt, abs, floor, ceil, round, sec, csc, cot. Supported constants: pi, e.

Example Data Table

Function	Order	Point	Expected Result	Reason
x^4 + 2*x^3 - x	2	2	72	Second derivative is 12x^2 + 12x.
exp(x)	5	1	2.718282	Every derivative of exp(x) remains exp(x).
sin(x)	4	0	0	The fourth derivative returns sin(x).

Formula Used

This page uses a numerical central finite-difference approach with Fornberg weights. It estimates the selected higher derivative from nearby function values around the evaluation point.

f^(n)(x0) ≈ Σ [ wj × f(x0 + jh) ]

Here, n is the derivative order, h is the step size, and wj are the finite-difference weights generated for the chosen stencil.

An error indicator is also shown by comparing the derivative computed with h and with h/2. A smaller gap often suggests a more stable estimate.

How to Use This Calculator

Enter a supported function expression using one variable, such as x^5 or sin(x)*exp(x).
Choose the derivative order you want to estimate.
Provide the point where the derivative should be evaluated.
Adjust the step size and stencil radius for stability and detail.
Set the graph range and graph point count.
Press the calculate button to see the result above the form.
Review the graph, local stencil table, and estimated error.
Download the graph summary as CSV or PDF when needed.

Frequently Asked Questions

1. Does this calculator return symbolic derivative formulas?

No. It estimates higher derivatives numerically at chosen points and across a graph range. It is designed for practical evaluation, trend checking, and exportable numerical results.

2. Which function types are supported?

It supports common algebraic, exponential, logarithmic, hyperbolic, and trigonometric functions, including sec, csc, and cot. Use explicit multiplication and parentheses for reliable parsing.

3. Why does step size matter?

A large step size can smooth away detail. A very small one can amplify rounding noise. The best value depends on function shape, derivative order, and the selected evaluation point.

4. What does stencil radius control?

Stencil radius controls how many nearby sample points are used. A larger stencil can improve accuracy for smooth functions, but it may also become sensitive near steep or discontinuous regions.

5. Why might results fail near corners or discontinuities?

Higher derivatives require local smoothness. Functions with sharp corners, jumps, singularities, or invalid domains can produce unstable values, missing graph points, or evaluation errors.

6. What does the estimated error mean?

It compares two numerical estimates built with different step sizes. It is a practical stability indicator, not a formal proof of exact accuracy.

7. What is included in the CSV export?

The CSV includes summary metrics and graph-ready data columns for x, the original function, and the selected higher derivative across the chosen interval.

8. How can I improve numerical stability?

Start with moderate step sizes, keep the graph range sensible, avoid invalid domains, and compare nearby settings. Extremely high orders usually need careful tuning and smoother functions.