# Basics of mathematical analysis.

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derivative of a function f (x) at a specific point x0 is the boundary function of the ratio of growth to the growth of the argument, provided that x is to be 0, and the boundary is.Derivative generally denoted by a prime, sometimes via point or via a differential.Often the entry is derived across the border leads to confusion, since such a representation is rarely used.

function which has a derivative at a certain point x0, is called differentiable at this point.Suppose, D1 - a set of points in which the function f is differentiated.To each one of the numbers x, belonging to D f '(x), we obtain a function with domain designation D1.This function is derivative of y = f (x).It is denoted: f '(x).

In addition, derivatives are widely used in physics and engineering.Consider a simple example.The material point moves on the coordinate directly to do with the law of motion is given, that is, the coordinate x of this point is a known function of x (t).During the time interval from t0 to t0 + t equals the displacement of the point x (t0 + t) -x (t0) = x, and an average speed v (t) equal to x / t.

Sometimes the character of the motion is presented, so that at small time intervals the average speed is not changed, meaning that the movement with a greater degree of accuracy is considered to be uniform.Alternatively, the average speed if t0 be absolutely accurate to a certain value, which is called the instantaneous velocity v (t0) of this point at a time t0.It is believed that the instantaneous speed v (t) is known for any differentiated function x (t), at what v (t) is equal to x '(t).Simply put, the speed - a derivative of coordinates with respect to time.

Instant speed has both positive and negative values, as well as the value of 0. If it is at a certain time interval (t1; t2) is positive, then the point moves in the same direction, that is, the coordinate x (t) increases withtime, and when v (t) is negative, then the coordinate x (t) decreases.

In more complex cases, the point moves in the plane or in space.Then the rate - a vector quantity, and defines each of the components of the vector v (t).

Similarly, we can compare with the acceleration of the point.Speed ​​is a function of time, ie v = v (t).A derivative of such a function - an acceleration of motion: a = v '(t).That is, it turns out that the derivative of the velocity with respect to time is acceleration.

Suppose y = f (x) - any differentiated function.Then we can consider the motion of a point on the coordinate axis, which is due by the law x = f (t).Mechanical maintenance of the derivative gives the opportunity to provide a clear interpretation of the theory of differential calculus.

How to find the derivative?Finding the derivative of a function is called its differentiation.

hover examples of how to find the derivative of the function:

derivative of a constant function is zero;derivative of the function y = x is equal to unity.

And how to find the derivative of the fraction?To do this, consider the following material:

For any x0 & lt; & gt; 0 we have

y / x = -1 / x0 * (x + x)

There are a few rules of how to find the derivative.Namely:

If the functions A and B are differentiated point x0, then their sum is differentiated point: (A + B) '= A' + B '.Simply put, the derivative of a sum equal to the sum of derivatives.If the function is differentiated to some point, then it must increment to zero when following the argument to zero gain.

If the functions A and B are differentiated at the point x0, then their product is differentiated at: (A * B) '= A'B + AB'.(The values ​​of functions and their derivatives are calculated at the point x0).If the function A (x) is differentiated point x0, and C - a constant function CA then differentiated at this point and (CA) '= CA'.That is, a constant factor taken outside the sign of the derivative.

If the functions A and B differentiated x0, the function B is not equal to zero, then their relationship as differentiated at: (A / B) '= (A'B-AB') / B * B.