Defining the Derivative
AP Calculus BC Topic 2 7:52 English narration · English + 中文 subtitles burned in
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Transcript
Draw a journey as a curve.
把一段行程画成曲线。
Pick two points, join them with a straight line, and its slope gives the average rate over an interval.
取两点，用一条直线把它们连起来，这条线的斜率给出整段区间上的平均变化率。
But we want the rate at one exact moment.
但我们想要的是某一个确切时刻的变化率。
So keep the first point fixed and slide the second point in, closer and closer.
于是让第一个点固定，把第二个点滑近，越来越近。
The line pivots, and finally rests on the tangent at that single point.
这条线不断转动，最后落在那一点的切线上。
Its slope is the derivative — the definition of the derivative.
它的斜率就是导数。
This unit defines the derivative at a point, then builds the rules that let us differentiate almost anything.
本单元先定义某一点处的导数，然后建立起让我们几乎能对任何函数求导的法则。
Let's begin.
让我们开始吧。
Start with the average rate of change.
先从平均变化率开始。
Take two points on the curve; the line joining them is a secant, and its slope is the change in output over the change in input.
在曲线上取两点；连接它们的直线叫割线， 它的斜率就是输出的变化除以输入的变化。
That expression is called the difference quotient.
这个式子叫做差商。
It comes in two equivalent forms — one using a step of size h from the point, the other using two separate x-values.
它有两种等价的写法——一种用从该点出发、步长为 h 的形式，另一种用两个不同的 x 值。
Both measure the same average rate.
两者衡量的是同一个平均变化率。
Now the key step.
现在是关键一步。
Let the step shrink toward zero.
让步长缩向零。
The secant pivots, and settles onto the tangent line.
割线不断转动，最后落在切线上。
The slope it settles on is the instantaneous rate of change — the derivative at that point.
它最终停在的那个斜率，就是瞬时变化率——也就是该点的导数。
Formally: the derivative at a is the limit of the difference quotient, as the step size goes to zero.
正式地说：a 处的导数，是当步长趋于零时差商的极限。
When that limit exists, we say the function is differentiable there.
当这个极限存在时，我们就说函数在那里可导。
You will meet three notations for the derivative — d y d x, f prime of x, and y prime.
你会看到导数有三种写法——d y d x、f 撇 x，以及 y 撇。
They all mean the same thing, so read whichever appears.
它们的意思完全相同， 看到哪一种就读哪一种。
And note: once we let the point vary, the derivative becomes a new function, giving the slope at every input.
并且注意：一旦让那个点自由变动，导数就变成一个新的函数， 在每个输入处给出斜率。
Geometric meaning: that slope is the tangent's gradient, so you can write the tangent line immediately with the point-slope form.
因为那个斜率就是切线的斜率，你可以立刻用点斜式写出切线方程。
Often the exam gives no formula at all — you are given only a table of values.
考试常常根本不给公式——只给你一张数值表。
Then estimate the derivative with a difference quotient over a small interval around the point, using the closest values on each side.
这时就用该点附近一个小区间上的差商来估计导数， 并取两侧最接近的值。
In a real-world model you must also attach the units: the output unit, per input unit.
在实际模型中，你还必须带上单位：每一个输入单位对应的输出单位。
That unit line is worth a mark on its own.
那一行单位本身就值一分。
The exam skill appears almost every year.
这项考试技能几乎每年都出现。
A question such as approximate M prime of seven point five using the average rate of change of M over five to ten is asking for one difference quotient.
题目如果说：用 M 在五到十上的平均变化率， 去近似 M 撇七点五， 问的就是一个差商。
Show the setup: M of ten minus M of five, all over ten minus five.
把式子写出来：M 的十减去 M 的五，再除以十减五。
Full credit needs the numbers plugged in and the correct units: output units per input unit, because these come from real-world models.
满分需要代入数字，还要写对单位： 每一个输入单位对应的输出单位， 因为这些来自真实模型。
Differentiability and continuity are linked, and the exam tests the link.
可导性与连续性是相连的，而考试就爱考这个联系。
Differentiability implies continuity: if a function is differentiable at a point, then it is continuous there.
如果一个函数在某点可导，那么它在那里连续。
But the reverse is false — a continuous function can still fail to be differentiable, in two ways.
但反过来不成立——一个连续函数仍然可能不可导，有两种情形。
At a corner, like the absolute-value graph, the slope from the left and the slope from the right disagree.
在尖点处，比如绝对值图象，左边的斜率和右边的斜率不一致。
At a vertical tangent, the slope is infinite.
在垂直切线处，斜率是无穷大。
Use the contrapositive: not continuous means not differentiable.
用逆否命题：不连续就一定不可导。
Higher-order derivatives are just derivatives of derivatives — the second derivative is the rate of change of the rate of change.
高阶导数不过是导数的导数——二阶导数就是变化率的变化率。
From here we use rules, not limits.
从这里开始我们用法则，不再用极限。
The most important is the power rule: bring the power down to the front, and lower the power by one.
最重要的是幂法则：把幂次拿到前面来，并把幂次减一。
It holds for any real power — whole numbers, negative powers, and roots, once you rewrite them as powers.
它对任何实数幂都成立——整数、负幂，以及先改写成幂的根式。
Add three more rules and you can differentiate term by term: a constant gives zero, a constant multiple comes along unchanged, and sums and differences split. Together these differentiate any polynomial term by term.
再加上另外三条法则，你就能逐项求导：常数的导数是零，常数倍数原样保留，和与差可以拆开。
These are the derivatives of cos x, sin x, e to the x, and ln x.
这些就是 cos x、sin x、e 的 x 次方和 ln x 的导数。
Four building blocks you must know by heart.
有四个基本导数你必须背熟。
The derivative of cosine is negative sine — watch that minus sign, it is the one students drop.
余弦的导数是负的正弦——注意那个负号，这正是学生最爱丢的。
And the derivative of sine is cosine, with no minus.
而正弦的导数是余弦，没有负号。
The exponential function is its own derivative, unchanged.
指数函数的导数就是它自己，保持不变。
And the derivative of the natural logarithm is one over x.
自然对数的导数是 x 分之一。
Memorise these four; everything else is built from them.
把这四个背下来；其余的一切都由它们搭建而成。
Also watch for a limit that is really a derivative — the exam codes it LIM.
还要留意那种其实就是导数的极限——考试记作 LIM。
If you recognize the difference quotient of a known function, just evaluate that derivative at the point.
若你认出某个已知函数的差商， 直接算该点处的导数即可。
Sometimes a limit is secretly the definition of a known derivative.
有时一个极限其实就是某个已知导数的定义。
If you recognise the difference quotient for a function whose derivative you know, just evaluate that derivative at the point.
若你认出一个你知道导数的函数的差商， 直接在该点求那个导数即可。
For example, the limit as h goes to zero of sine of pi over two plus h, minus one, over h, is the derivative of sine at pi over two, which is cosine of pi over two, which is zero.
例如，h 趋于零时， 正弦的二分之派加 h，减去一，再除以 h， 就是正弦在二分之派处的导数， 也就是余弦的二分之派，等于零。
Do not expand the sine.
不要把正弦展开。
Name the derivative.
说出那个导数。
What about a product of two functions?
那么两个函数的乘积呢？
You cannot just multiply the two derivatives — that is simply wrong.
你不能只是把两个导数相乘——那完全是错的。
Use the product rule: the derivative of the first, times the second, plus the first, times the derivative of the second.
要用乘积法则：第一个的导数乘以第二个，加上第一个乘以第二个的导数。
For x squared times e to the x, that gives two x e to the x, plus x squared e to the x.
对 x 平方 乘以 e 的 x 次方，就得到 二 x e 的 x 次方，加上 x 平方 e 的 x 次方。
For a quotient, use the quotient rule: the bottom times the derivative of the top, minus the top times the derivative of the bottom.
对于商，就用商法则：分母乘以分子的导数，减去分子乘以分母的导数。
The order matters, because of that minus sign — swap the two terms and you flip the sign and lose the mark.
次序很重要，因为有那个负号——把两项交换，符号就反了，分数也就丢了。
All of that is then divided by the bottom squared.
然后把这一切除以分母的平方。
Now the other four trigonometric derivatives — and you do not memorise them separately.
现在讲另外四个三角函数的导数——你不必分别去背它们。
Instead, rewrite each one with identities as a quotient and apply the rule you just learned.
而是把每一个改写成商，再套用你刚学的法则。
Tangent becomes sine over cosine, and the quotient rule gives one over cosine squared, the secant squared.
正切写成正弦除以余弦， 商法则就给出余弦平方分之一，也就是正割的平方。
All four follow the same way: cotangent gives minus cosecant squared, secant gives secant times tangent, and cosecant gives minus cosecant times cotangent.
四个都照这个办法来： 余切给出负的余割平方，正割给出正割乘正切，余割给出负的余割乘余切。
Notice the pattern — every co-function picks up a minus sign.
注意这个规律——每一个"余"函数都带一个负号。
Let's finish with one.
我们用一道题收尾。
Differentiate sine x, over x.
对 正弦 x 除以 x 求导。
First read the structure: it is a quotient, so the quotient rule applies.
先读结构：它是一个商，所以用商法则。
The top is sine x, whose derivative is cosine x.
分子是 正弦 x，它的导数是 余弦 x。
The bottom is x, whose derivative is one.
分母是 x，它的导数是一。
Now combine: bottom times the derivative of the top, minus top times the derivative of the bottom, over the bottom squared.
现在合起来： 分母乘以分子的导数，减去分子乘以分母的导数，再除以分母的平方。
That gives x cosine x, minus sine x, all over x squared.
这就给出 x 余弦 x，减去 正弦 x，全部除以 x 平方。
Keep that order in the numerator — swapping it flips the sign.
分子里的次序要保持——交换它就会反号。
Before you go, three marks to protect.
结束之前，三个要守住的分。
First, the derivative is the slope of the tangent — the limit of the secant slope.
第一，导数是切线的斜率——也就是割线斜率的极限。
Second, differentiable implies continuous, but never the reverse: a corner is continuous with no derivative.
第二，可导必连续，但反过来绝不成立：尖点是连续的，却没有导数。
Third, in the quotient rule, keep the order in the numerator — bottom times the derivative of the top comes first.
第三，商法则中要保持分子的次序——分母乘以分子的导数放在前面。
Get these right, and this topic is yours.
把这些做对，这个专题就是你的了。