Let’s suggest a function y=f(x) that is defined on the interval (a,b). Choose a point x on the interval (a,b), and another point x+∆x of this interval. ∆x is an increment of the function argument at the point x.
When x is fixed, ∆y is a function of ∆x. This is an increment of function y=f(x), in the point x.
The equation above is also a function of ∆x.
The derivative of a function =f(x) in the point x in the following limit: .
In mathematics derivative marks f'(x) or y'(x).
In physics the following derivative uses: y ̇(x) or .
Derivative of some functions:
1. Function of a constant y=c, where c is a number.
2. Power function , n belongs to a real numbers unity. Then,
If exists, it names the right derivative of the function y=f(x) in the point x. It’s mark is
In the same way the left derivative of function is y=f(x) in the point x is: .
Let us look at the function y=|x|. In the x=0 we have:
Let us discuss the function z=f(x;y), the function of two variables x and y. If we fix for example, variable y, then we have the function f of one variable x. The Derivative of z=f(x;y) by variable is called partial derivative, and marks zx’. In the same way partial derivative zy’ is defined by the variable y.
The magnitude is an instaneous velocity of a part on the moment of time x.
In case of a random function y=f(x), its derivative f’ (x) is the velocity of change of variable y corresponding to change of variable x.
Let’s take a function y=f(x), where the unity of points ((f(x);x),x∈X) where X is a unity of points on this line. Let’s put points M(x,f(x)) and N(x+∆x,f(x+∆x)) on the line. The string MN is a secant in relation to the function on the graph. The angle φ(∆x) between the secant MN and 0X axis. Let’s suggest .
If , so line l with slope k=tan , which goes through the point M(x,f(x)) is a tangent to the function y=f(x) in the M point.
Also line l is a critical position of the secant MN, with . So we can say that the tangent to the function f(x) in the point M(x,f(x)) is a critical position of its secant MN, when .
If function y=f(x) has a derivative f'(x) in the point x, so y=f(x) has a tangent in the point M(x,f(x)), and its slope is equal to f'(x).
Let’s have a look on the MNP triangle:
By the tangence definition, tangence to the graph exists in the point M(x,f(x)).
And hence, .
Tangence equation to the y=f(x), in the point has the following form: