钙衍生物(CALCULUS DERIVATIVES)

After derivative theory posts, we will start to see some of the applications that make this technique one of the most important in mathematics and therefore at machine learning.

在派生理论发表后，我们将开始看到使该技术成为数学中最重要的应用之一，因此成为机器学习中最重要的应用之一。

函数的最大值和最小值 (Maximum and Minimum of a function)

The theorem of the local maximum that we already introduced, can be extrapolated to the whole function domain.

我们已经介绍的局部极大值定理可以外推到整个函数域。

Let f be a function and A a set of numbers contained in the domain of f. A point x in A is a maximum point for f on A if f(x)≥f(y) for every y in A. The number f(x) itself is called the maximum value of f on A.

设f是一个函数， A是包含在f的域中的一组数字。 在A A点x是对于f的最大点上的一个，如果F(X)≥f(Y)，用于在每Y个。 f(x)本身称为A上f的最大值。

Notice that there can be multiple maximum points for the same function at distinct x values. The minimum point definition is obtained inverting the previous definition, ergo changing f(x)≥f(y) for f(x)≤f(y).

请注意，在不同的x值下，同一函数可以有多个最大点。 通过颠倒先前的定义获得最小点定义，对于f(x)≤f(y)进行ergo更改f(x)≥f(y) 。

最高和最低点的导数 (Derivatives at maximum and minimum points)

As you can expect, maximum and minimum points will always be a change in the derivative of the function, that allows us to demonstrate that:

正如你可以预期，最大我的妈妈和最低点永远是在衍生功能的变化，使我们能够证明：

Let f be any function defined on (a,b). If f is a maximum or a minimum point for f on (a,b), and f is differentiable at x, then f’(x)=0.

令f为(a，b)上定义的任何函数。 如果f是最大或对于f的最小点上(A，B)，和f是在x处可微，则f'(X)= 0。

局部最大值和最小值 (Local maximums and minimums)

Let f be a function, and A a set of numbers contained in the domain of f. A point x in A is a local maximum[minumum] point for f on A of there is some δ > 0 such that x is a maximum[minumum] point for f on A ⋂ (x-δ,x+δ ).

令f为一个函数， A为f的域中包含的一组数字。 在A A点x是局部最大值[minumum]上的一个点与F的存在一些δ> 0，使得x是一个最大[minumum]点对A⋂F(X-δ中，x +δ)。

关键点 (Critical points)

Not all x that makes f’(x) = 0 will be maximums or minimums, they are types of critical points:

并非所有使f'(x)= 0的x都是最大值或最小值，它们是临界点的类型：

A critical point of a function f is a number x such that f’(x)=0. The number f(x) is called a critical value of f.

临界点 函数f是一个数字x ，使得f'(x)= 0 。 数f(x)称为f的临界值。

In order to find the maximum and minimum of f, we have to check the following values:

为了找到f的最大值和最小值，我们必须检查以下值：

• The critical points of f in [a,b].

  f在[a，b]中的临界点。

• The endpoints, a and b.

  端点a和b 。

• Points x in [a,b] such that f is not differentiable at.

  在[a，b]中指向x ，使得f不可微分。

临界点检测的一些重要定理 (Some important theorems for critical point detection)

罗尔定理(The Rolle’s Theorem)

If f is continuous on [a,b] and differentiable on (a,b), and f(a)=f(b), then there is a number x in (a,b) such that f’(x)=0.

如果f在[a，b]上是连续的并且在(a，b)上是可微的，并且f(a)= f(b) ，则在(a，b)中存在一个数x ，使得f'(x)= 0 。

This can lead to a constant function or to a function that changes the gradient between both values, so f’(x)=0.

这可能导致常数函数或导致两个值之间的梯度改变的函数，因此f'(x)= 0 。

均值定理 (The mean value theorem)

Rolle theorem allows us to demonstrate the following:

罗尔定理允许我们证明以下内容：

Iff is continuous on [a,b] and differentiable on (a,b), then there is a number x in (a,b) such that

如果f是在[a，b]的连续可微的(A，B)，则存在在数x(A，B)，使得

分类关键点 (Classifying critical points)

增减功能(Increasing and decreasing functions)

A function f is increasing on an interval if f(a)<f(b) whenever a and be are two numbers in the interval with a<b. The function f is decreasing on an interval if f(a) > f(b) for all a and b in the interval with a<b.

如果a和be是区间a <b中的两个数字，则当f(a)<f(b)时，函数f在区间上增大。 如果对于区间a <b中的所有a和b的f(a)> f(b) ，函数f在区间上减小。

临界点分类的二阶导数 (Second derivatives for critical point classification)

Now we know how to find a critical point with the first derivative and check the type of them using the left and right derivative values, we can use the second derivative to skip the lateral limit calculations.

现在我们知道了如何使用一阶导数找到临界点并使用左右导数值检查它们的类型，我们可以使用二阶导数来跳过横向极限计算。

Suppose f’(a)=0. If f’’(a)>0, then f has a local minimum at a; if f’’(a)<0 the nf has a local maximum at a.

假设f'(a)= 0 。 如果f'(A)> 0，则f具有在局部最小值; 如果f'(a)中<0 NF具有在局部最大值。

简化的两个强定理 (Two strong theorems for simplification)

柯西中值定理(The Cauchy Mean value theorem)

If f and g are continuous on [a,b] and diffrentiable on (a,b), then there is a number x such that [f(b)-f(a)]g’(x)=[g(b)-g(a)]f’(x).

如果f和g在[a，b]上是连续的并且在(a，b)上是可微的，则存在一个数字x使得[f(b)-f(a)] g'(x)= [g(b )-g(a)] f'(x) 。

If g(b)≠g(a), and g’(x)≠0, this can be written as:

如果g(b)≠g(a)和g'(x)≠0 ，则可以写成：

洛皮塔尔的统治 (L’Hôpital’s rule)

结论 (Conclusion)

In this post, we introduced how to use derivatives to find local maximums and minimums, they allow us to find possible solutions to our cost function optimization. This will allow us to determine the gradient easier.

在本文中，我们介绍了如何使用导数查找局部最大值和最小值，它们使我们能够找到成本函数优化的可能解决方案。 这将使我们更容易确定梯度。

This is the twenty-seventh post of my particular #100daysofML, I will be publishing the advances of this challenge at GitHub, Twitter, and Medium (Adrià Serra).

这是我特别＃的第二十七届后100daysofML，我会发布在GitHub上，Twitter和中型企业(这一挑战的进步阿德里亚塞拉)。

https://twitter.com/CrunchyML

https://twitter.com/CrunchyML