The basic rules of differentiation used in mathematics

To begin with, it is worth remembering what a differential is and what a mathematical meaning it carries.

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

In turn, by the definition of the derivative of the function y '= lim dx-0 (dy / dx), and by the definition of the limit - dy / dx = x' + α, where the parameter α is an infinitesimal mathematical value.

Therefore, both sides of the expression should be multiplied by dx, which finally gives dy = y '* dx + α * dx, where dx is an infinitesimal change of the argument, (α * dx) is a value that can be neglected, then dy is the increment Function, and (y * dx) - the main part of the increment or differential.

A differential of a function is the product of the derivative of a function by the differential of the argument.

Now we should consider the basic rules of differentiation, which are often used in mathematical analysis.

Theorem. The derivative of the sum is equal to the sum of the derivatives obtained from the summands: (a + c) '= a' + c '.

Similarly, this rule will also apply to finding the derivative of the difference.
A consequence of this differentiation rule is the assertion that the derivative of a certain number of terms is equal to the sum of the derivatives obtained from these summands.

For example, if it is necessary to find the derivative of the expression (a + c-k) ', then the result is the expression a' + c'-k '.

Theorem. The derivative of the product of mathematical functions differentiate 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: (a * c) '= a * c' + a '* s. The corollary of the theorem is the conclusion that the constant factor in the derivative product can be taken as 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 derivative of the expression (2a3) ', then the result is the answer: 2 * (a3)' = 2 * 3 * a2 = 6 * a2.

Theorem. The derivative of the ratio of functions is the ratio between the difference of the numerator derivative multiplied by the denominator and the numerator multiplied by the denominator derivative and the denominator square.

Mathematically, the theorem will be written as follows: (a / c) '= (a' * c-a * c ') / c ² .

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

Theorem. Suppose given a function y = φ (χ), where χ = c (m), then the function y with respect to the variable τ is called complex.

Thus, in mathematical analysis, the derivative of a complex function is treated 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.

With the regular use of this table, derivatives are easily remembered. The remaining derivatives of complex functions can be found by applying the rules for differentiating functions that have been stated in theorems and corollaries to them.