For starters, it's worth remembering what a differential is and what mathematical meaning it carries.
The differential of a function is the product of the derivative of a function from an argument and the differential of the argument itself. Mathematically, this concept can be written as an expression: dy=y'dx.
In turn, according to the definition of the derivative of the function, the equality y'=lim dx-0(dy/dx) is true, and according to the definition of the limit, the expression dy/dx=x'+α, where the parameter α is an infinitesimal mathematical value.
Therefore, both parts of the expression should be multiplied by dx, which ultimately gives dy=y'dx+αdx, where dx is an infinitesimal change in the argument, (αdx) is a value that can be neglected, then dy is the increment of the function, and (ydx) is the main part of the increment or differential.
The differential of a function is the product of the derivative of a function and the differential of the argument.
Now it's worth considering the basic rules of differentiation, which are quite often used in mathematical analysis.
Theorem. The derivative of the sum is equal to the sum of the derivatives obtained from the terms: (a+c)'=a'+c'.
Similarlythis rule will also apply to finding the derivative of the difference.
The consequence of this differentiation rule is the statement that the derivative of a certain number of terms is equal to the sum of the derivatives obtained from these terms.
For example, if you need to find the derivative of the expression (a+c-k)', then the result will be the expression a'+c'-k'.
Theorem. The derivative of the product of mathematical functions that are differentiable at a point is equal to the sum consisting of the product of the first factor by the derivative of the second and the product of the second factor by the derivative of the first.
Mathematically, the theorem will be written as follows: (ac)'=ac'+a'c. A consequence of the theorem is the conclusion that the constant factor in the derivative of the product can be taken out of the derivative of the function.
In the form of an algebraic expression, this rule will be written as follows: (ac)'=ac', where a=const.
For example, if you need to find the derivative of the expression (2a3)', then the result will be the answer: 2(a3)'=23a2=6a2.
Theorem. The derivative of the ratio of functions is equal to the ratio between the difference between the derivative of the numerator multiplied by the denominator and the numerator multiplied by the derivative of the denominator and the square of the denominator.
Mathematically, the theorem will be written as follows: (a/c)'=(a'c-ac')/c^{2}.
In conclusion, it is necessary to consider the rules for differentiating complex functions.
Theorem. Let the function y \u003d f (x), where x \u003d c (t), then the function y, with respect toto the variable m, is called complex.
Thus, in mathematical analysis, the derivative of a complex function is interpreted as the derivative of the function itself, multiplied by the derivative of its subfunction. For convenience, the rules for differentiating complex functions are presented in the form of a table.
f(x) |
f^{'}(x) |
(1/s)' | -(1/s^{2})s' |
(а^{с})' | a^{c}(ln a)c' |
(е^{с})' | e^{c}c' |
(ln s)' | (1/s)s' |
(log a^{c})' | 1/(сlg a)c' |
(sin c)' | cos ss' |
(cos c)' | -sin ss' |
With regular use of this table, derivatives are easy to remember. The remaining derivatives of complex functions can be found by applying the rules for differentiating functions that were stated in the theorems and corollaries to them.