## Description

This app calculates the analytic formula of the first derivative of a given function. Derivative represents a rate of change of a function with respect to an independent variable. This app enables to differentiate a function that depends on more than one independent variable (called here a multi-parametric case) by considering these variables separately. To do so, one must declare a name of independent variable with respect to which a given function is differentiated. During differentiation, the rest of variables is kept constant. The derivative found on the basis of the obtained analytical formula can be compared with the derivative found numerically. The obtained results can then be depicted in plots. In order to plot a multi-parametric case, all remaining independent variables must be set by the user. To make this app more didactic, it is possible to track and practice intermediate stages of searching for the analytic derivative. All these steps can be performed by the user himself. In this way, the final formula of the derivative sought will become more understandable.

## Expression To Differentiation

In order to properly construct the expression to differentiate, follow the instructions below.

### Math Operators

Please remember about operators between expression factors, e.g. 1.5x should be written as: 1.5 * x
subtraction: -
multiplication: *
division: /
exponentiation: ^.

### The use of parentheses

Starting parenthesis: (
The closing parenthesis :)

### Real Numbers

Notation of real numbers: e.g. 2.05, 3.86, 1.8, 8.5 and the like.

### Math Functions

natural logarithm: log(x)
logarithm to the base 'a': log_a(x), where 'a' is a positive real number
sine: sin(x)
cоsine: cos(x)
tangent: tan(x)
cotangent: ctan(x)
inverse sine: asin(x)
inverse cosine: acos(x)
inverse tangent: atan(x)
inverse cotangent: actan(x)
hyperbolic sine: sinh(x)
hyperbolic cosine: cosh(x)
hyperbolic tangent: tanh(x)
hyperbolic cotangent: ctanh(x)
inverse hyperbolic sine: asinh(x)
inverse hyperbolic cosine: acosh(x)
inverse hyperbolic tangent: atanh(x)
inverse hyperbolic cotangent: actanh(x)
power function: x^(a), where a is a real number
exponential function: a^(x), where 'a' is a positive real number (e.g. a = e = 2.71828..)

## Differentiation Variable

The differentiation variable is defined as the independent variable after which the first derivative of a function will be calculated. The default is 'x', but any name is allowed except where it starts 'w'.

## Expression Parameters

The number of parameters is defined as the number of independent variables or numeric constants in the expression without the differentiation variable. This number will be important when plotting the chart and when studying intermediate stages.

The number of parameters must be less than 100 because a higher value results in an impractically large number of parameters to enter.

## Solution

The obtained solution is additionally presented in a simplified form, more convenient for analysis. In addition, in the obtained function, coloring or highlighting of parentheses is available.

## Singularities

Some of the detected singularities may in fact be removable singularities. They can arise due to the structure of an expression e.g. sin(x)/x with singularity at zero. In fact, there is no singularity at all.

The localization of detected singularities depends on the calculation step. To achieve a higher resolution around the singularity point set the plot range in close neighbourhood of this point.

## Intermediate Stages

The user can independently study the intermediate steps of finding the derivative of the function that led to the obtained solution. These stages result from the division of the complex function into internal functions. Finding the first derivative of a function is shown as a derivative chain of internal functions, where the last element is one of the elementary functions.
The rules of finding the first derivative of a function can be found at this link.

## Plot

The application allows you to draw a graph of a function and its further editing. The resulting image can be saved and then shared as an email or MMS attachment.
Specifically, these features are available when editing the plot: zoom, fonts and colors, titles and legend editing, marker and line types, axis marker format.

## Availability

### Apple iOS Mobile Devices

#### AppClip (iOS version 14.0 or higher)

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