Derivative Calculator • With Steps!

Switch light/dark mode Derivative Calculator

Calculate derivatives online — with steps and graphing!

Also check the Integral Calculator!Calculadora de Derivadas en españolAbleitungsrechner auf Deutsch About Help Examples Options Practice

The Derivative Calculator lets you calculate derivatives of functions online — for free!

Our calculator allows you to check your solutions to calculus exercises. It helps you practice by showing you the full working (step by step differentiation).

The Derivative Calculator supports computing first, second, …, fifth derivatives as well as differentiating functions with many variables (partial derivatives), implicit differentiation and calculating roots/zeros. You can also check your answers! Interactive graphs/plots help visualize and better understand the functions.

For more about how to use the Derivative Calculator, go to "Help". Also see "Examples".

And now: Happy differentiating!

Enter the function you want to differentiate into the Derivative Calculator. Skip the f(x)= part! The Derivative Calculator will show you a graphical version of your input while you type. Make sure that it shows exactly what you want. Use parentheses, if necessary, e.g. a/(b+c). Write decimal fractions with a period instead of a comma, e.g. 3.141.

In "Examples" you will find some of the functions that are most frequently entered into the Derivative Calculator.

When you're done entering your function, click "Go!", and the Derivative Calculator will show the result below.

In "Options" you can set the differentiation variable and the order (first, second, … derivative). You can also choose whether to show the steps and enable expression simplification.

Click an example to enter it into the Derivative Calculator (the current input will be deleted).

$2x$ $x^{2}$ $\frac{1}{x}$ $\sqrt{x}$ $\sqrt[3]{x}$ $\frac{1}{\sqrt{x}}$ $\ln(x)$ $\log_{10}(x)$ $\ln^{2}(x)$ $\ln(x^{2} + 1)$ $x \ln(x)$ $\frac{x}{\ln(x)}$ $\mathrm{e}^x$ $\mathrm{e}^{-x}$ $2^x$ $\mathrm{e}^{-x^{2}}$ $x^x$ $x\mathrm{e}^x$ $x\mathrm{e}^{-x}$ $\sin(x)$ $\cos(x)$ $\tan(x)$ $\sec(x)$ $\sin^2(x)$ $\cos^2(x)$ $x \sin(x)$ $x \cos(x)$ $\cos(x) \sin(x)$ $\ln(\cos(x))$ $\arcsin(x)$ $\arctan(x)$ $|x|$ $\frac{1}{x + 1}$ $\frac{1}{1 - x^{2}}$ $\frac{x}{x^{2} + 1}$ $\operatorname{f}(x) \operatorname{g}(x)$ $\frac{\operatorname{f}(x)}{\operatorname{g}(x)}$ $\operatorname{f}(\operatorname{g}(x))$ $\operatorname{f}^{\operatorname{g}(x)}(x)$

Configure the Derivative Calculator:

Differentiation variable:
Differentiate how many times?	1 2 3 4 5
Simplify expressions?
Simplify all roots?(√x² becomes x, not |x|)
Use complex domain ℂ?
Keep decimals?
Show calculation steps?
Calculate roots/zeros?
Implicit differentiation?
Dependent variable:(will be treated as a function)

The practice problem generator allows you to generate as many random exercises as you want.

You find some configuration options and a proposed problem below. You can accept it (then it's input into the calculator) or generate a new one.

Inverse trigonometric/hyperbolic functions
Hyperbolic functions
Cosecant, secant and cotangent

Accept problem Next problem

Calculate the Derivative of …Enter your own Answer:

Exit "check answer" mode CLR + – × ÷ ^ √ f(x) π ( ) √ 3√ 4√ n√ You can also input:• sqrt(…)• root(n, …) ln log10 logn exp ex abs |x| sin cos tan csc sec cot arcsin sin-1 arccos cos-1 arctan tan-1 arccsc csc-1 arcsec sec-1 arccot cot-1 sinh cosh tanh csch sech coth arsinh sinh-1 arcosh cosh-1 artanh tanh-1 arcsch csch-1 arsech sech-1 arcoth coth-1 f f ' f '' g g ' g '' Si Ci Shi Chi Ei E1 En li erf erfc erfi W Lambert Γ Gamma Β Beta ψn Polygamma S Fresnel C Fresnel Lin Polylog π 3.141… e 2.718… i √-1 φGR 1.618… γEM 0.577… α β γ δ ϵ ε ζ η θ ϑ ι κ ϰ λ μ ν ξ ο ϖ ρ ϱ σ ς τ υ ϕ φ χ ψ ω Γ Δ Θ Λ Ξ Π Σ Υ Ψ Ω You can also input:• pi for π, alpha for α, …• upper_… for uppercase letters

This will be calculated:

Loading … please wait!This will take a few seconds.

Use parentheses, if necessary. Also see "Examples". Change differentiation variable and order in "Options".

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Result

Above, enter the function to derive. Differentiation variable and more can be changed in "Options". Click "Go!" to start the derivative calculation. The result will be shown further below.

How the Derivative Calculator Works

For those with a technical background, the following section explains how the Derivative Calculator works.

First, a parser analyzes the mathematical function. It transforms it into a form that is better understandable by a computer, namely a tree (see figure below). In doing this, the Derivative Calculator has to respect the order of operations. A specialty in mathematical expressions is that the multiplication sign can be left out sometimes, for example we write 5x instead of 5*x. The Derivative Calculator has to detect these cases and insert the multiplication sign.

The parser is implemented in JavaScript, based on the Shunting-yard algorithm, and can run directly in the browser. This allows for quick feedback while typing by transforming the tree into LaTeX code. MathJax takes care of displaying it in the browser.

When the "Go!" button is clicked, the Derivative Calculator sends the mathematical function and the settings (differentiation variable and order) to the server, where it is analyzed again. This time, the function gets transformed into a form that can be understood by the computer algebra system Maxima.

Maxima takes care of actually computing the derivative of the mathematical function. Like any computer algebra system, it applies a number of rules to simplify the function and calculate the derivatives according to the commonly known differentiation rules. Maxima's output is transformed to LaTeX again and is then presented to the user.

Displaying the steps of calculation is a bit more involved, because the Derivative Calculator can't completely depend on Maxima for this task. Instead, the derivatives have to be calculated manually step by step. The rules of differentiation (product rule, quotient rule, chain rule, …) have been implemented in JavaScript code. There is also a table of derivative functions for the trigonometric functions and the square root, logarithm and exponential function. In each calculation step, one differentiation operation is carried out or rewritten. For example, constant factors are pulled out of differentiation operations and sums are split up (sum rule). This, and general simplifications, is done by Maxima. For each calculated derivative, the LaTeX representations of the resulting mathematical expressions are tagged in the HTML code so that highlighting is possible.

The "Check answer" feature has to solve the difficult task of determining whether two mathematical expressions are equivalent. Their difference is computed and simplified as far as possible using Maxima. For example, this involves writing trigonometric/hyperbolic functions in their exponential forms. If it can be shown that the difference simplifies to zero, the task is solved. Otherwise, a probabilistic algorithm is applied that evaluates and compares both functions at randomly chosen places.

The interactive function graphs are computed in the browser and displayed within a canvas element (HTML5). For each function to be graphed, the calculator creates a JavaScript function, which is then evaluated in small steps in order to draw the graph. While graphing, singularities (e.g. poles) are detected and treated specially. The gesture control is implemented using Hammer.js.

If you have any questions or ideas for improvements to the Derivative Calculator, don't hesitate to write me an e-mail.

© David Scherfgen 2025 — all rights reserved.

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