Table of contents
1. Algebraic derivation
Suppose we have a square whose initial side length is X, then the area S1=x²
Then the side length of the square increases by △x, at this time the area S2=(x+△x)²
The area of change is △s=(x+△x)²- x²=2x△x+(△x)²
It can be observed that when △x is smaller (△x)² will be closer to 0 than 2x△x, that is to say, when △x is small, we can approximate that
△s=2x△x
Carefully observe the above formula, this 2X is actually the derivative of the square of x. At this time, do we understand why the derivative can describe the speed of the changing trend?
2. Geometry derivation
Geometric derivation is actually geometric meaning.
An image will be made of the above functions to understand the geometric meaning of the differential:
K: A line connecting two points AB
L: Tangent to point A
Next calculate △y:
There are two calculation methods for Δy:
①Function substitute value, subtracting the beginning from the end:
△y=(x+△x)²-x²=x²+2△x+（△x）²-x²
=2x△x+(△x)²
②Calculation by slope:
∵△y/△x=k
∴△y=k△X
It has been deduced before that the (derivative) slope of x² is 2x,
That is to say, the slope of each point on the function changes with x.
The slope at A is: 2x
The slope at B is: 2(x+△x)
Observe the two straight lines K and L in the figure:
The slope of the K line is the slope between two points x and x+△x.
Line L is the tangent to point A, and the slope of line L is the slope of point A.
Whether it is observing the graph or verifying it according to the function of the slope above, it can be seen that:
When △ x is smaller, x+ △ x is closer to point x :
Both the K line and the function graph are infinitely close to the L line, and the three coincide at point A.
mean what?
It means that when △ x is small enough, the L line can replace the function image and the K line for calculation , that is, the slope between the two points A and B can be approximately equal to the slope of the point A, so:
△y=2X△X
In summary, the two △y are obtained by two methods above:
Direct substitution to find:
△y=2x△x+(△x)²
Find from the slope:
△y=2X△X
It can be found that the difference between the exact value obtained by direct substitution and the approximate value obtained by slope approximation is:
△X², when the change is small enough and △X approaches 0, △X² can be omitted naturally.
In fact, it can also be strictly deduced from the image that △X² is 0, which can save:
BC is the △y corresponding to the direct substitute value,
CD is △y obtained with infinite approach
The BD section is actually △X²
△X is small enough, the third line approaches infinitely, and the BC five insurances approach CD. BD approaches 0 infinitely, which means that △X² approaches 0 infinitely, so it can be omitted naturally.
Summarize:
To summarize the final result of the above derivation process:
f(X), when △X->0:
△y=f’(X)△x
Replace △ with a symbol d
dy= f’(X)dx
d is called the differential sign.
(In fact, the transformation of the formula here can also prove that the derivative is the quotient of the differential—the differential quotient—f'(X)=dy/dx)
3. Summary
To summarize what differentiation really is:
In human terms, it means infinite approximation, cutting a big thing into small enough things, so that some small quantities before being divided can be ignored, and the calculation focuses on the general rather than the details.
In non-human language:
Local linearization of nonlinear functions, local linearization of curves.
Let me explain a little bit about this non-human language:
In the three-line infinite approximation, it can be seen that the straight line of the tangent function is used to replace the original curve function image for calculation, and the original nonlinear function is approximately linearized. This provides a solution to many difficult-to-calculate scenarios in engineering.