The derivative of some function f (x) at a specific point x0 is the boundary of the ratio of the growth of the function to the growth of the argument, provided that x follows to 0, and the boundary exists. The derivative is usually indicated by a prime, sometimes with a dot or through a differential. Often, recording derivatively 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 called differentiable at such a point. Suppose D1 is the set of points at which the function f is differentiated. Assigning to each number a number x belonging to D f '(x), we obtain a function with the notation D1. This function is the derivative of y = f (x). It is denoted as follows: f '(x).
In addition, the derivative is widely used in physics and technology. Consider the simplest example. The material point moves along the coordinate straight, with which the law of motion is specified, that is, the x coordinate of this point is the well-known function x (t). Over the time interval from t0 to t0 + t, the movement of the point is x (t0 + t) -x (t0) = x, and its average speed v (t) is x / t.
Sometimes the nature of the movement is presented in such a way that the average speed does not change for small periods of time, meaning that the movement is considered uniform with a greater degree of accuracy. Or the value of the average speed, if t0 follows to some absolutely exact value, which is called the instantaneous speed v (t0) of this point at a particular time t0. It is believed that the instantaneous velocity v (t) is known for any differentiated function x (t), and v (t) will be equal to x '(t). Simply put, speed is a derivative of a time coordinate.
The instantaneous velocity has both positive and negative values, as well as 0. If it is positive in a certain time interval (t1; t2), then the point moves in the same direction, that is, the coordinate x (t) increases with time, and if v (t) is negative, then the x (t) coordinate 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, it can be compared with the acceleration of a point. Speed is a function of time, i.e. v = v (t). And the derivative of such a function is acceleration of motion: a = v '(t). That is, it turns out that the derivative of speed with respect to time is an acceleration.
Suppose y = f (x) is any differentiated function. Then we can consider the motion of the 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 differential calculus theorems .
How to find a derivative? Finding the derivative of a function is called its differentiation.
Here are some 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 unity.
And how to find a 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 growth follows to zero when the argument grows to 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 constant, then the function CA is differentiated at this point and (CA) '= CA'. That is, such a constant factor is taken out of the derivative sign.
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.