The basic rules of differentiation, applied mathematics

For a start it is worth remembering that such differential and a mathematical meaning it carries.

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

In turn, by definition, the derivative of the equality y '= lim dx-0 (dy / dx), and to determine the limit - the expression dy / dx = x' + α, where the parameter α is infinitesimal mathematical quantity.

Consequently, both parts of the expression is multiplied by dx, which eventually gives dy = y '* dx + α * dx, where dx - is an infinitesimal change in the argument, (α * dx) - the value of which can be ignored,then dy - increment of the function, and (y * dx) - the main part of the increment or differential.

differential of the function is the product of the derivative function on the differential argument.

now is to consider the basic rules of differentiation, which are often used in mathematical analysis.

Theorem. derivative amount equal to the sum of the products obtained from components: (a + c) = a '+ c'.

Similarly, this rule will be valid for the derivative of the difference.
consequence danogo rules of differentiation is the assertion that the derivative of a number of terms is equal to the sum of the products obtained by these terms.

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

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

mathematical theorem is written as follows: (a * c) '= a * a' + a * s.The consequence of the theorem is the conclusion that the constant factor in the derived product can be taken out of the derivative of the function.

as an algebraic expression, this rule will be recorded as follows: (a * a) = a * s', where a = const.

For example, if you want to find the derivative of the expression (2a3) ', then the result will be an answer: * 2 (a3) ​​= 2 * 3 * 6 * a2 = a2.

Theorem. derivative relations function is the ratio between the difference of the derivative of the numerator multiplied by the denominator and the numerator is multiplied by the square of the derivative of the denominator and the denominator.

mathematical theorem is written as follows: (a / c) '= (A' *, with a * c ') / s2.

In conclusion, it is necessary to consider the rules of differentiation of complex functions.

Theorem.Let a fuktsii y = f (x), where x = s (t), then the function y with respect to the variable T called complex.

Thus, in the mathematical analysis of the derivative of a composite function is treated as a derivative of the function multiplied by the derivative of its sub-functions.For your convenience the rule for differentiating composite functions are in the form of a table.

With regular use of derivatives in this table are easy to remember.The rest of the derivatives of complex functions can be found, if we apply the rules of differentiation of functions that have been stated in the theorems and corollaries to them.