Application of the first principle of derivatives:
We have already seen that the derivative of a function y = f(x) by the first principle of differentiation is given by:
|f ' (x)||=||dy|
|=|| lim |
h -> 0
|f (x + h) - f(x)|
In other words, we first find the difference quotient
and then find its limit as h -> 0.
Now let's apply this principle to find the derivatives of some elementary functions.
Find the derivative of the following functions by the first principle:
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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.
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 systemMaxima.
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.
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.
If you have any questions or ideas for improvements to the Derivative Calculator, don't hesitate to write me an e-mail.