The differential of the function is the principal (linear by ) part of function increment. To understand this definition, consider the following figure.
The figure shows the graph of the function and its tangent at the point . Let's give the function's argument some increment , then the function will also get some increment . The value is called the differential of the function . From the figure, it follows, that is equal to the increment of the ordinate of the tangent, drawn at the point to the function . This is why the differential is called linear part of the increment of the function, i.e. the increment of the ordinate of the tangent.
From the figure, it follows, that the slope angle of the tangent , which it forms with the positive direction of -axis and are equal. In addition, the tangent slope angle of the tangent is equal to the value of the derivative of the function at the point of tangency:
From triangle, it follows, that
Thus, the differential of the function is expressed by the following formula:
Consider one more point: the figure above shows, that , and . And, the smaller , the smaller the contribution to the value of makes the value of . I.e., for sufficiently small values, we can assume, that . This relationship allows one to calculate the approximate value of the function at point , if its value at the point is known.
A higher-order differential (for instance, the order of ) is defined as the differential from an -th order differential:
For example, the second-order differential is calculated as follows:
Similarly, one can obtain a formula for calculating the -th order differential:
where is the -th derivative of the function with respect to variable.
A few words should be spoken about calculating the differential of the many variables function. In this case the differential is called the total differential and for the function depending on -variables is defined by the formula:
Expressions for higher order differentials of the many variables function can be obtained from the general formula:
Generally, to raise the sum to -th power, one should use Newton's Binomial theorem. Consider the process of obtaining the second order total differential formula of two variables function:
Our online calculator is able to find differentials of different orders for any single or many variables functions with step by step solution.