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: limx→0yx.

In mathematics derivative marks f'(x) or y'(x).
In physics the following derivative uses: y ̇(x) or dydx.

Derivative of some functions:
1. Function of a constant y=c, where c is a number.


limx→0 yx1=0,c‘=0

2. Power function y=xn, n belongs to a real numbers unity. Then, y=f(x+∆x)-f(x)=(x+∆x)nxn=xn+nxn-1x+n(n-1)x(n-2)(∆x)22+⋯+(∆x)nxn=nxn-1x+o(x), when x0, yx=nxn-1+o(∆x)x nxn-1, where x  0, then (xn)’=nxn-1, where n belongs to the natural numbers unity

3. y=sin xy=sin (x+∆x)−sin x=2sin x2*cos (x+x2) = 2 (x2+o(x)*cos (x+x2), x  0yx=(1+o(x)x*cos (x+x2), x 0Finally (sin x)’=cos x.

One-sided derivatives

If limx→+0 (yx)1 exists, it names the right derivative of the function y=f(x) in the point x. It’s mark is fr‘ (x).

In the same way the left derivative of function is y=f(x) in the point x is: limx→-0 (yx)1= fl‘ (x).

Let us look at the function y=|x|. In the x=0 we have:

y=y(0+x)y(0)=x,x>0x,x<0,therefore yx=+1,x>01,x<0
Partial derivatives

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.

Physical and geometrical meaning of derivatives. Physical meaning of derivative
Assume that x is time, and y=f(x) is the coordinate of a point on the X0Y plane at the moment of time x.

yx=f(x+∆x)-f(x)x. is an average velocity of this point during a period of time from x till ∆x.
The magnitude limx→0 yx=f‘ (x)=v(x) 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.

Geometric meaning of derivatives
Let’s assume the string l at the X0Y coordinate system. To align the 0X axis of a coordinate system with string l we have to turn it to the angle π2<α<π2. The k=tan⁡ α is the slope of the string l in the coordinate system X0Y.

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 x  0.

Then ∆y=f(x+∆x)-f(x).

If limx→0 φ(Δx)=φ0, so line l with slope k=tan⁡ φ0, 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 x  0. 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 x  0.

Theorem 1
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:

tan φ(Δx)=yx, in this way φ(∆x)=tan-1yx.

limx→0yx=f‘ (x), additionally tan-1t is an incontinuous function.

limx→0 φ(∆x)=limx→0 tan-1yx=tan-1f‘ (x).Here φ0=limx→0 φ(∆x)=tan-1 f‘ (x).

By the tangence definition, tangence to the graph exists in the point M(x,f(x)).
And hence, k=tan φ0=f(x).

Tangence equation to the y=f(x), in the point M(x0,f(x0)) has the following form: yf(x0)=f‘ (x0)(xx0).

Differentiability, differential of a function and integral

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