Interactive Function Derivative Calculator

Function f(x)

Variable

Order n

Method

Point x₀

Interval

Points on plot

Step h for numeric

—

Simplified expression to copy:

—

Value at x₀:

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x	f(x)	f⁽ⁿ⁾(x)

Think of this derivative calculator as your fast lab for exploring derivatives. It gives you the n-th derivative formula when possible and a reliable numeric estimate if the symbolic route fails. Use it to get a clear formula, a plug-and-play numeric value at a point, graphs across a range, and quick diagnostics to catch common mistakes.

How the calculator works

The tool offers two main modes. The symbolic mode tries to produce an exact expression for the n-th derivative using algebraic differentiation and simplification. If the symbolic engine succeeds you get a neat formula ready to copy and the numeric value at the target x0. The numeric mode builds an approximation using central finite differences applied recursively for higher order derivatives. Numeric mode is fast and robust for expressions that are too large or procedural for symbolic treatment.

After you hit Compute the app attempts symbolic differentiation when requested, compiles the function for numeric evaluation, plots f(x) and the derivative f⁽ⁿ⁾(x), fills a table of sampled values, and reports the derivative at your chosen x0. If symbolic differentiation fails the app falls back to numeric and shows a clear message so you know what happened.

Here are the practical details you care about most: pick the variable name, choose the order n as a nonnegative integer, set the evaluation point x0, choose an interval and number of sample points for a chart, and set the finite difference step h for numeric mode. Smaller h increases accuracy for low n but can introduce round off errors for extremely small h. For higher n increase h moderately to reduce cancellation errors.

Quick examples and expected output

Example 1. Input function sin(x) * exp(x), order n = 2, x0 = 1. Symbolic output is f”(x) = 2 e^x cos x. Numeric evaluation at x0 = 1 gives roughly 2.937388 and matches the symbolic result.

Example 2. Input function x^3 + 2*x, order n = 1, x0 = 2. Symbolic derivative is 3 x^2 + 2 and the numeric value at x0 = 2 is 14.

Example 3. If the input uses conditional logic or external functions that the symbolic engine cannot reduce, the interface reports symbolic failure and computes a numeric approximation instead. Numeric mode will always try to deliver a reliable value when the function is evaluable at the sampled points.

Essential math rules implemented

Rule	Formula
Constant	$\frac{d}{dx} c = 0$
Power rule	$\frac{d}{dx} x^{m} = m x^{m-1}$
Linearity	$\frac{d}{dx}[a f + b g] = a f’ + b g’$
Product rule	$(f g)’ = f’ g + f g’$
Quotient rule	$\left(\frac{f}{g}\right)’ = \frac{f’ g – f g’}{g^{2}}$
Chain rule	$(f\circ g)'(x) = f'(g(x)) \cdot g'(x)$
Exponential	$(e^{u})’ = u’ e^{u}$
Logarithm	$(\ln x)’ = 1/x$
Trig	$(\sin x)’=\cos x,\; (\cos x)’=-\sin x$

Numeric method and practical guidance

The numeric algorithm uses central finite difference. For first derivative the formula is

$$
f'(x)\approx \frac{f(x+h)-f(x-h)}{2h}
$$

For second derivative a standard second order formula is

\[
f”(x) \approx
\]
\[
\frac{f(x+h) – 2 f(x) + f(x-h)}{h^2}
\]

The calculator applies a recursive central formula for higher order derivatives so that

\[
D^n f(x) \approx
\]
\[
\frac{D^{n-1}f(x+h) – D^{n-1}f(x-h)}{2h}
\]

Recommendations for choosing h: for low orders n = 1 or 2 choose h near 1e-4 to 1e-6. For higher orders raise h toward 1e-3 or 1e-2 to limit catastrophic cancellation. If numeric results swing wildly when you change h by a small factor, increase sample density and compare with symbolic when available.

Use numeric mode when symbolic differentiation fails or when function contains piecewise code.
Check that x0 and nearby x plus or minus h are inside the function domain for sqrt log and rational expressions.
Limit n to a reasonable value. Numeric accuracy degrades rapidly beyond n of eight to twelve depending on the function.

Troubleshooting quick list

If you get wrong or noisy answers check these things in the order listed

Syntax errors. Use caret for power, write sin(x) and exp(x) and avoid stray characters.
Domain errors. For log and sqrt make sure sampled points stay positive when required.
Step h too small. Increase h to reduce round off error for higher derivatives.
Order n too large. Try symbolic if possible or reduce n and analyze lower derivatives first.

Test functions to validate behavior

Try these to compare symbolic and numeric outputs

sin(x) * exp(x) for 1st and 2nd derivatives
x^5 + 3 x^2 – 7 for polynomial checks
log(x) * x^2 with x positive to test domain handling
abs(x) to see non differentiable point behavior at zero
sin(10 x) with high order derivatives to reveal numeric instability

When symbolic fails

Symbolic differentiation can stumble on procedural code, conditionals, or very large expressions. That is normal. The app will display a polite message and switch to numeric mode automatically. You can then tune h and sample points to validate the numeric estimate. When possible compare numeric output to the simplified symbolic expression to estimate approximation error.

📝 The calculator produces a TeX rendering of symbolic results when available, and offers a plain text expression for copy and paste. Save your input expression and settings so you can reproduce or vary the run quickly. If you need a publication quality formula copy the TeX output to your paper or slides.

Closing note

This derivative calculator combines symbolic algebra and reliable numeric schemes so you get the best of both worlds. Start in auto mode, inspect the symbolic formula when it appears, and use numeric mode to handle edge cases and validate results. If something looks off tweak the finite difference step and keep an eye on the domain of evaluation.