y=f(x)=x2, 求f'(x). 直线的斜率k=(y1-y0)/(x1-x0)=((x+d)2 - x2) / (x+d - d) = (2xd + d2) / d = 2x + d = 2x
d非常小，无限接近于0，可以被忽略掉。x很大时xd能不能被忽略掉？反正xd/d总等于x。长得像近似解却能得到精确解？这道理本小白也不明白，大白才明白。
对x3求导时，3x2+3xd+d2，x只是可能很大，比如10987，并不是无穷大，d则是无穷小。我们不考虑f'(∞)。无穷大乘无穷小是不定型极限。
sinx求导挺好玩的 - Fun_with_Words - 博客园
郎X平(大意)"偏微分方程我学不明白，你们更学不明白"，一堆犯贱喊好的(反赚九百万依然能蹦跶)。本小白和他不一样——咱们不会的程度是一样的。:-)
周毓麟先生最主要的研究领域是偏微分方程。偏微分方程的求解比代数方程困难得多，很多情况下甚至连有没有解都很难判断。[链接] [链接2]
斜率相等的直线有无数条。x2的导数derivative是2x不是说y=2x是y=x2的切线。以x=3为例，此时切线的斜率为6，且过(3,9)点，方程是(y-9)=6(x-3), 即y=6x-9.
切线帮助理解导数，导数不(仅仅)是为了研究切线。以y=x3为例，导数是3x2. x=0时，斜率为0的x轴是x3的切线吗？
y=x的导数是y=1，直线没有切线，还是切线就是它本身？The tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
推导复合函数求导公式的窗户纸是分子分母都乘以g(x+d) - g(x). 可以用(x2+1)2或x6练习下，把后者看成x2和x3的复合函数。
现在我们来看3维。z=x描述一个平面。求偏导数时把不被求的看作常数即可，z对x求偏导是1，对y求偏导是0。y轴在z=x这个平面里(z=x=0, y任意)，它们之间的夹角是0°。x轴和该平面的夹角是45°。偏导对应的是tangent，与二维里的斜率类似。当z对y的偏导数是0时，y变z不变。
z=x2+y是一个曲面的方程。它对x的偏导是2x，对y的偏导是1。
下山时如何减少z？改变x还是y？改变哪个效果更明显？比如上图中我们想从右上角到z=0。可以先减少x走到谷底，再顺着缓坡滑下来。也可以先只减少y，最后顺着陡坡滑下来。x和y还可以同时变。If a hill is defined as a height function over a plane h(x,y), the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.
斜率和速率都是数没有方向。梯度和速度都有方向。(1,0)和(1,0,0)分别是2维和3维空间里x轴上的单位向量。2x乘(1,0)得到(2x,0)，代表一堆都在x轴上但大小不同的向量。想改变y=x2的值，只能改变x，所有的梯度(2x, 0)都在x轴上。想改变z=x2+y的值，除了横着和竖着走，还可以斜着走，梯度(2x, 1)都在xOy平面内，Origin代表原点。坐标(2,3)处的梯度是(4,1)，可把起点为(0,0)，终点为(4,1)的向量平移为起点为(2,3)。(1,2)表示点，<1,2>表示向量不好吗? 还有超标量：1是标量，(1,2,3)+(2,3,4)得到(3,5,7)，它们是一堆数不是1个数，不关心它们是不是向量有没有方向。
梯度下降是顺着梯度的方向下降，不是减少梯度的模(长度)。长度是没有方向的标量啊，和上段中度和率的用法不统一。
点积对应的是cosine. 线性代数里可以对两个万维向量求夹角的余弦值，图没法画，计算方法则像是把for(i=0;i<2;i++)变成了(i=0;i<10000;i++). The dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean [欧几里得] geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called "the" inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus [向量微积分] and differential geometry [微分几何]. 偏微分还用于深度学习(多层神经网络)。In ULSI (Ultra Large Scale Integration), the number of gate circuits [门电路] per chip is >=50,000. 搞人工智能的会不会以后也缺形容词？Deep learning is a subset of machine learning, which is essentially a neural network with >=3 layers. stackexchange：
- "Deep" is a marketing term: you can therefore use it whenever you need to market your multi-layered neural network.
- In 2014 the "very deep" VGG netowrks Simonyan et al. (2014) consist of 16+ hidden layers.
- In 2016 the "extremely deep" residual networks He et al. (2016) consist of 50 up to 1,000+ hidden layers.
导数和导函数不同，把x代入导函数可求出导数。Differentiation is the action of computing a derivative. The differential [微分] of a function represents a change in the linearization of a function. Linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion [泰勒展开] around the point of interest. 一阶展开f'，二阶f''，依次类推。A differential equation [微分方程] is an equation that relates one or more functions and their derivatives. The problem of determining the area [面积] of plane figures was a major motivation for the historical development of calculus [微积分]. 积分符号是拉长的花体s, s代表sum.
partial:
1. of or forming a part; not complete 部分的; 不完全的
2. showing too much favour to one person or side; biased 偏心
3. having a strong liking for sb/sth, 偏爱
我对x和y一点偏爱/偏见也没有。我只是觉得先把一个当常量，先干一部分工作比较简单。即我认为partial derivative应该翻译为部分导数。
How To Pronounce Partial derivative symbol?
∂不是希腊字母。The character ∂ is a stylized cursive d mainly used as a mathematical symbol, usually to denote a partial derivative such as ∂z/∂x. 数学符号∂ (HTML元素: ∂ 或 ∂, Unicode: U+2202) 可能是d的一种变体，主要用于表示偏导数，读作z对x的偏导数。
https://math.stackexchange.com/questions/120504/how-do-you-pronounce-partial-derivatives
http://www.uefap.com/speaking/symbols/symbols.htm [很长，偏导数在后面]
Delta(uppercase Δ, lowercase δ) is the fourth letter of the Greek alphabet. The nabla symbol (∇), also known as the inverted pyramid, inverted delta, inverted triangle, or inverted py, is the upside-down greek letter delta (Δ). The nabla symbol ∇ is the vector differential operator, also called del.
上文中three-dimensional，在我看来算4维，不过像0-based和1-based，不影响理解。i, j, k分别是x, y, z轴上的单位向量(1, 0, 0), (0, 1, 0)和(0, 0, 1)。轻率地猜测：
- 向量微积分难；但如果熟悉了这些符号，第一章应该基本看的懂。
- 非数学系毕业的个别网上大拿，其实也就第二、三章的水平：公式像套话，抄呗。