Derivation calculator

Calculate deductions online
— with calculation and graphs!

Also test the Integral calculator!
Derivative Calculator in English
Calculadora de Derivadas en espanol

The derivative calculator calculates online derivatives of any function – free of charge!

Check your calculus homework with this online calculator. It helps you to learn by showing you the complete calculation path.

The derivation calculator can calculate the first, second.., Calculate fifth derivative. Derivatives of functions with several variables (partial derivatives), implicit derivatives as well as the calculation of zeros are no problem. You can also check your solutions! Interactive function graphs make it easier to understand.

More about the operation of the derivative calculator can be found under"Help", or look at the examples.

And now: Happy deduction!

Enter the function you want to derive into the input field. Let thereby"f(x) =" away. The derivative calculator shows you a graphical version of the input. Make sure that they are exactly that shows what you mean. If necessary, set parentheses to achieve this, z. B. "a/(b+c)". Write decimal fractions with a period instead of a comma, i.e. z. B. "3.141".

Under"Examples" you can see which functions are supported and how to use them.

When you have entered your function, click on"Go!". After a short calculation time the result is then displayed below.

Under"Options" you can Derivative variable and the Order set (1., 2., … derivative) and the Show calculation path and the Simplify expressions on or. turn off.

Click takes over the example into the derivative calculator. Hover your mouse over it to display the text.

How the derivative calculator should work?

Derivative variable: Calculate to how many derivatives? Simplify expressions? Simplify all roots?
(√ x² becomes x instead of |x|) Complex range of values (ℂ)? Keep decimal fractions? Show calculation path? Calculate zeros? Implicit derivative? Dependent variable:
(is treated like function)

With the task generator you can generate as many random exercises as you want.

Below you will find settings and a suggested task. You can accept it (then it will be entered into the computer) or generate a new one.

Trigonometric/hyperbolic inverse functions Hyperbolic functions Cosecants, secants and cotangents

Accept task Next task

What do you want to derive? Enter your own solution:

The following is calculated:

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

Not what you mean? Set parentheses! Set derivative variable and order in"Options".

Recommend derivative calculator

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Derivation calculator


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ResultEnter the function to be derived above. Derivative variable and more you can find in"Options" change. Click"Go!", to start the calculation of the derivative. The result is shown below.

How the derivative calculator works

For the technically interested user, here is a brief explanation of how the derivative calculator works.

The entered mathematical function is first analyzed by a parser. The parser transforms the mathematical function into a structure that is easier for the computer to process, namely a tree (see image below). The derivation calculator must take into account the ranking of different operators (z. B. the "dot before dash-Rule). There is also a special feature of mathematical expressions: The multiplication sign is often omitted, z. B. write "5x instead of "5*x. The derivation calculator must recognize these cases and add the multiplication sign.

The parser is programmed in JavaScript (based on the shunting-yard algorithm) and can therefore be executed directly in the user’s browser. This allows immediate feedback while the mathematical function is still being entered. For this purpose, a LaTeX representation of the function is generated from the tree generated by the parser. MathJax takes care of the display in the browser.

Will the "Lot!"-If you click the button, the derivation calculator sends the mathematical function in its original form together with the settings (derivation variable and number of derivatives) to the server. There the function is analyzed again. This time, however, the function is converted into a different form so that it can be understood by the computer algebra system Maxima.

Maxima takes over the calculation of the derivatives. Like any computer algebra system, it applies a set of rules to simplify the function and derive it according to the commonly known derivation rules – just like you learn it in math class. The output of Maxima is then converted back into LaTeX form and presented to the user.

Displaying the calculation path is a bit more complicated. Here, the calculator cannot completely rely on maxima, but must perform the derivations itself step by step. For this purpose all derivation rules (product rule, quotient rule, chain rule, …) were implemented in JavaScript code. For the trigonometric functions, the root, logarithm and exponential function, the corresponding derivatives are stored in a table. In each calculation step, a derivation is performed or rewritten, z. B. constant factors are written in front of the derivative and sums in derivatives are separated (sum rule). The latter as well as general simplifications of the functions are taken over by Maxima. For each derivation performed, the LaTeX codes of the resulting expressions are specially marked in the HTML code, so that later color highlighting is possible.

The "solution check-Function has the difficult task of determining for two mathematical expressions whether they are equivalent. For this purpose, their difference is formed and simplified as much as possible with the help of maxima. Hereby z. B. trigonometric/hyperbolic functions into their exponential form. If it can be shown in this way that the difference is zero, then the problem is solved. Otherwise, a probabilistic algorithm is applied, which evaluates and compares the functions at randomly selected points.

The interactive function graphs are calculated in the browser and displayed in a canvas element (HTML5). For this purpose, the computer generates a JavaScript function from the entered function and the calculated derivatives in each case, which is finally evaluated in small steps to draw the graph. When drawing the function graph, definition gaps such as z. B. traced pole places and treated them specially. The gesture control is with hammer.js implemented.

Do you have any questions or suggestions for improvement to the derivation calculator? Did it help you with studying or exam preparation? You are welcome to send me an e-mail. I appreciate any feedback!

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