Mathematical basis of artificial intelligence series of articles

  • 1. Mathematical basis of artificial intelligence —- derivative

  • 2. Mathematical basis of artificial intelligence —- matrix

  • 3. Mathematical basis of artificial intelligence —- Linear second-order approximation


The study of artificial intelligence for math requirements or needs certain skills, whether it’s algorithm or involve the noun concept, is built on the basis of mathematical model for training of learning, so it is necessary to involve the mathematics knowledge are understood and comb again, to change the thinking from the traditional way of programming.

What is introduced here is the derivative of unary function (scalar field), and the derivative of multivariate function (vector or multi-dimensional matrix field) will be introduced later, because multivariate function requires knowledge related to vector and matrix, and then the derivative of multivariate function will be introduced in detail after the correlation between vector and matrix

A, derivative

Definition 1.

The derivative of the function f'(x0) is the derivative of the function f(x) at x0 and the slope of the tangent line of the function f(x) at x0, which is represented by point P here, as shown in the figure below:

2. Derivation process of derivation

We know from high school that the formula for the slope of a function is y-y0 = m(x – x0), where m is the slope of the function. So how do we figure out the slope or the derivative.

The coordinates of points P0 and Q are: P0(x0, f(x0)), Q(x0+ δ x, f(x0+ δ x)), y-y0 = m(x-x0), m = (y-y0)/(x-x0), m = δ f / δ x. To find the slope of P0, we need to introduce the concept of a limit, the slope, or the derivative, is as follows (Q and P0 overlap as delta x approaches 0) :

3. Take the derivative example

Example a

So according to this formula, for example, f of x is equal to 1 over x, what’s the derivative with respect to x0?

Example 2

After the derivative of function 1/x is obtained, we can solve an interesting problem. We can find the area of the triangle surrounded by the intersection of the tangent line at the point P of function F (x) = 1/x and the area of triangle AOB as shown in the following figure:

AO

Example 3

Since f(x) = 1/x (x to the -1), we can take the derivative of f(x) = x^n, or we can take the derivative of f(x) = x^n, as follows:

Example 4

The derivative of the trigonometric function f(x) = sinx, f ‘(x) = (sinx)’, using the derivation formula of the door, can be obtained:

The derivative of the cosine function f(x) =cosx, f ‘(x) =cosx ‘:

Second, higher derivatives

A higher order derivative is, if you take a derivative of a function once you call it the first derivative, if you take a derivative of a function again you call it the second derivative, if you take a derivative of a function again you call it the third derivative, if you take a derivative of a function again you call it the fourth derivative, if you take a derivative of a function n times you call it the NTH derivative, these are all higher order derivatives. So here’s an example, the NTH derivative of the function f of x is equal to x to the n, and what do I do? Newton used f prime of x for the first derivative, And Leibniz used d dx of x to the n for the derivative and dx to the n for the first derivative, d dx of d dx of x to the n for the second derivative and d squared x to the n, and d to the n x to the n

3. Common derivative formulas

Derivative knowledge is first introduced to this, about the derivation of the four operations, there are a lot of information on the Internet, you can find the relevant derivation rules on the Internet, all changes can use the derivation of the second subtitle of the “derivation formula” to calculate derivation. I hope this article is helpful for you to recall high school derivative and differential related content.




Mathematical basis of artificial intelligence series of articles

  • 1. Mathematical basis of artificial intelligence —- derivative

  • 2. Mathematical basis of artificial intelligence —- matrix

  • 3. Mathematical basis of artificial intelligence —- Linear second-order approximation