Fundamentals of mathematical analysis. How to find the derivative?

The derivative of some function f (x) at a specific point x0 is the boundary of the ratio of the increment of a function to the increment of an argument, provided that x follows to 0, and the boundary exists. The derivative is usually denoted by a prime, sometimes by a point or through a differential. Often, a record drawn across the border is misleading, since such a representation is used extremely rarely.

A function that has a derivative at a certain point x0 is said to be differentiable at such a point. Suppose that D1 is the set of points at which f is differentiated. By assigning to each number the number x belonging to D f '(x), we obtain a function with the domain of designation D1. This function is the derivative of y = f (x). It is denoted as f '(x).

In addition, the derivative is widely used in physics and engineering. Let's consider the simplest example. The material point moves along the coordinate axis directly, at what the law of motion is given, that is, the coordinate x of this point is the known function x (t). During the time interval from t0 to t0 + t, the displacement of the point is x (t0 + t) -x (t0) = x, and its average velocity v (t) is x / t.

Sometimes the character of the motion is represented in such a way that, for small time intervals, the average speed does not change, meaning that the motion is considered to be more uniform with a greater degree of accuracy. Or the value of the mean velocity, if t0 follows to some absolutely exact value, which is called the instantaneous velocity v (t0) of this point at a specific instant of time t0. It is assumed that the instantaneous velocity v (t) is known for any differentiated function x (t), with v (t) being equal to x '(t). Simply put, speed is the derivative of the time coordinate.

The instantaneous velocity has both positive and negative values, and also a value of 0. If it is positive for some time interval (t1; t2), then the point moves in the same direction, that is, the x (t) coordinate increases with time, and if V (t) is negative, then the coordinate x (t) decreases.

In more complex cases, the point moves in a plane or in space. Then the velocity is a vector quantity and determines each of the coordinates of the vector v (t).

Similarly, one can compare with the acceleration of the motion of a point. Speed is a function of time, that is, v = v (t). And the derivative of such a function is the acceleration of motion: a = v '(t). That is, it turns out that the derivative of the speed with respect to time is an acceleration.

Suppose that y = f (x) is any differentiate function. Then we can consider the motion of a material point along the coordinate line, which occurs behind the law x = f (t). The mechanical content of the derivative makes it possible to present a visual interpretation of the theorems of differential calculus.

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

We will give examples of how to find the derived function:

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

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

For any x0 <0 we have

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

There are several rules for finding a derivative. Namely:

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

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

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