For example, this involves writing trigonometric/hyperbolic functions in their exponential forms. Their difference is computed and simplified as far as possible using Maxima. The "Check answer" feature has to solve the difficult task of determining whether two mathematical expressions are equivalent.
#Derivative of log 2x code#
For each calculated derivative, the LaTeX representations of the resulting mathematical expressions are tagged in the HTML code so that highlighting is possible. This, and general simplifications, is done by Maxima.
For example, constant factors are pulled out of differentiation operations and sums are split up (sum rule). In each calculation step, one differentiation operation is carried out or rewritten. There is also a table of derivative functions for the trigonometric functions and the square root, logarithm and exponential function. The rules of differentiation (product rule, quotient rule, chain rule, …) have been implemented in JavaScript code. Instead, the derivatives have to be calculated manually step by step. Maxima's output is transformed to LaTeX again and is then presented to the user.ĭisplaying the steps of calculation is a bit more involved, because the Derivative Calculator can't completely depend on Maxima for this task. 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 takes care of actually computing the derivative of the mathematical function. This time, the function gets transformed into a form that can be understood by the computer algebra system Maxima. 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. MathJax takes care of displaying it in the browser. This allows for quick feedback while typing by transforming the tree into LaTeX code. The parser is implemented in JavaScript, based on the Shunting-yard algorithm, and can run directly in the browser. The Derivative Calculator has to detect these cases and insert the multiplication sign. A specialty in mathematical expressions is that the multiplication sign can be left out sometimes, for example we write "5x" instead of "5*x". In doing this, the Derivative Calculator has to respect the order of operations. It transforms it into a form that is better understandable by a computer, namely a tree (see figure below). That derivative approaches 0, that is, becomes smaller.ī) when x is less than 1 and becomes smaller.For those with a technical background, the following section explains how the Derivative Calculator works.įirst, a parser analyzes the mathematical function. Calculate the derivative of lnĪccording to the rule for changing from base e to a different base a:Ī) when x is greater than 1 and becomes larger. When y = e u( x), then according to the chain rule:Įxample 4. The derivative of e with a functional exponent In the system of natural logarithms, in which e is the base, we have the simplest constant possible, namely 1. ( Lesson 39 of Algebra.) When we calculate that derivative below, we will see that that constant becomes ln a.
Where k is the constant of proportionality. Therefore, to say that the rate of growth is proportional to its size, is to say that the derivative of a x is proportional to a x. The more individuals there are, the more births there will be, and hence the greater the rate of change of the population - the number of births in each year.Īll exponential functions have the form a x, where a is the base. The bigger it is at any given time, the faster it's growing at that time. For we say that a quantity grows "exponentially" when it grows at a rate that is proportional to its size.
What does that imply? It implies the meaning of exponential growth.