The basic rules of differentiation used in mathematics

First of all, it is worth remembering what a differential is and what mathematical sense it carries.

The differential of a function is the product of the derivative of the function from the argument by the differential of the argument itself. Mathematically, this concept can be written as an expression: dy = y "* dx.

In turn, by definition, the derivativefunctions, the equality y "= lim dx-0 (dy / dx) is true, and by definition the limit is the expression dy / dx = x" + α, where the parameter α is an infinitely small mathematical quantity.

Therefore, both parts of the expression should be multiplied.on dx, which ultimately gives dy = y "* dx + α * dx, where dx is an infinitely small change in the argument, (α * dx) is a value that can be neglected, then dy is the increment of the function, and (y * dx ) - the main part of the increment or differential.

The differential of a function is the product of the derivative of the function and the differential of the argument.

Now it is necessary to consider 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 derived from the components: (a + c) "= a" + c ".

Similarly, this rule will act to find the derivative of the difference.
The consequence of this rule of differentiation 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 it is necessary to find a derivative of the expression (a + ck) ", then the result will be the expression a" + c "-k".

Theorem. Derivative product of mathematical functions,differentiable at a point, is equal to the sum consisting of the product of the first multiplier by the derivative of the second and the product of the second multiplier by the derivative of the first.

Математически теорема будет записана следующим in the following way: (a * c) "= a * c" + a "* c. The result of the theorem is the conclusion that the constant factor in the derivative of a 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: (a * c) "= a * c", where a = const.

For example, if it is necessary to find the derived expression (2a3) ", then the result will be the answer: 2 * (a3)" = 2 * 3 * a2 = 6 * a2.

Theorem. The derivative of the ratio of functions is equal to the ratio between the difference of 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" * s-a * s ") / s².

In conclusion, it is necessary to consider the rules for differentiating complex functions.

Theorem. Let the function y = φ (χ) be given, where x = c (m), then the function y, with respect to the variable m, is called complex.

So in mathematical analysisthe derivative of a complex function is treated as a derivative of the function itself, multiplied by the derivative of its subfunction. For convenience, the rules of differentiation of complex functions are presented in the form of a table.

With regular use of this tablederivatives are easily remembered. The remaining derivatives of complex functions can be found if we apply the rules of differentiation of functions that were set forth in theorems and consequences for them.