y=f(x)=x2, 求f'(x). 直線的斜率k=(y1-y0)/(x1-x0)=((x+d)2 - x2) / (x+d - d) = (2xd + d2) / d = 2x + d = 2x
d非常小，無限接近於0，可以被忽略掉。x很大時xd能不能被忽略掉？反正xd/d總等於x。長得像近似解卻能得到精確解？這道理本小白也不明白，大白才明白。
對x3求導時，3x2+3xd+d2，x只是可能很大，比如10987，並不是無窮大，d則是無窮小。我們不考慮f'(∞)。無窮大乘無窮小是不定型極限。
sinx求導挺好玩的 - Fun_with_Words - 博客園
郎X平(大意)"偏微分方程我學不明白，你們更學不明白"，一堆犯賤喊好的(反賺九百萬依然能蹦躂)。本小白和他不一樣——咱們不會的程度是一樣的。:-)
周毓麟先生最主要的研究領域是偏微分方程。偏微分方程的求解比代數方程困難得多，很多情況下甚至連有沒有解都很難判斷。[鏈接] [鏈接2]
斜率相等的直線有無數條。x2的導數derivative是2x不是說y=2x是y=x2的切線。以x=3為例，此時切線的斜率為6，且過(3,9)點，方程是(y-9)=6(x-3), 即y=6x-9.
切線幫助理解導數，導數不(僅僅)是為了研究切線。以y=x3為例，導數是3x2. x=0時，斜率為0的x軸是x3的切線嗎？
y=x的導數是y=1，直線沒有切線，還是切線就是它本身？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.
推導復合函數求導公式的窗戶紙是分子分母都乘以g(x+d) - g(x). 可以用(x2+1)2或x6練習下，把后者看成x2和x3的復合函數。
現在我們來看3維。z=x描述一個平面。求偏導數時把不被求的看作常數即可，z對x求偏導是1，對y求偏導是0。y軸在z=x這個平面里(z=x=0, y任意)，它們之間的夾角是0°。x軸和該平面的夾角是45°。偏導對應的是tangent，與二維里的斜率類似。當z對y的偏導數是0時，y變z不變。
z=x2+y是一個曲面的方程。它對x的偏導是2x，對y的偏導是1。
下山時如何減少z？改變x還是y？改變哪個效果更明顯？比如上圖中我們想從右上角到z=0。可以先減少x走到谷底，再順着緩坡滑下來。也可以先只減少y，最后順着陡坡滑下來。x和y還可以同時變。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.
斜率和速率都是數沒有方向。梯度和速度都有方向。(1,0)和(1,0,0)分別是2維和3維空間里x軸上的單位向量。2x乘(1,0)得到(2x,0)，代表一堆都在x軸上但大小不同的向量。想改變y=x2的值，只能改變x，所有的梯度(2x, 0)都在x軸上。想改變z=x2+y的值，除了橫着和豎着走，還可以斜着走，梯度(2x, 1)都在xOy平面內，Origin代表原點。坐標(2,3)處的梯度是(4,1)，可把起點為(0,0)，終點為(4,1)的向量平移為起點為(2,3)。(1,2)表示點，<1,2>表示向量不好嗎? 還有超標量：1是標量，(1,2,3)+(2,3,4)得到(3,5,7)，它們是一堆數不是1個數，不關心它們是不是向量有沒有方向。
梯度下降是順着梯度的方向下降，不是減少梯度的模(長度)。長度是沒有方向的標量啊，和上段中度和率的用法不統一。
點積對應的是cosine. 線性代數里可以對兩個萬維向量求夾角的余弦值，圖沒法畫，計算方法則像是把for(i=0;i<2;i++)變成了(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 [歐幾里得] 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 [向量微積分] and differential geometry [微分幾何]. 偏微分還用於深度學習(多層神經網絡)。In ULSI (Ultra Large Scale Integration), the number of gate circuits [門電路] per chip is >=50,000. 搞人工智能的會不會以后也缺形容詞？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.
導數和導函數不同，把x代入導函數可求出導數。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 [泰勒展開] around the point of interest. 一階展開f'，二階f''，依次類推。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 [面積] of plane figures was a major motivation for the historical development of calculus [微積分]. 積分符號是拉長的花體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, 偏愛
我對x和y一點偏愛/偏見也沒有。我只是覺得先把一個當常量，先干一部分工作比較簡單。即我認為partial derivative應該翻譯為部分導數。
How To Pronounce Partial derivative symbol?
∂不是希臘字母。The character ∂ is a stylized cursive d mainly used as a mathematical symbol, usually to denote a partial derivative such as ∂z/∂x. 數學符號∂ (HTML元素: ∂ 或 ∂, Unicode: U+2202) 可能是d的一種變體，主要用於表示偏導數，讀作z對x的偏導數。
https://math.stackexchange.com/questions/120504/how-do-you-pronounce-partial-derivatives
http://www.uefap.com/speaking/symbols/symbols.htm [很長，偏導數在后面]
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，在我看來算4維，不過像0-based和1-based，不影響理解。i, j, k分別是x, y, z軸上的單位向量(1, 0, 0), (0, 1, 0)和(0, 0, 1)。輕率地猜測：
- 向量微積分難；但如果熟悉了這些符號，第一章應該基本看的懂。
- 非數學系畢業的個別網上大拿，其實也就第二、三章的水平：公式像套話，抄唄。