Defining the Derivative
AP Calculus AB Topic 2 7:54 English narration · English + 中文 subtitles burned in
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Picture a car's journey as a curve.
把一辆车的行程画成一条曲线。
Over a whole interval, we can measure its average speed — the slope of the straight line joining two points on the curve.
在整段区间上，我们可以量出它的平均速度—— 也就是连接曲线上两点的直线的斜率。
But what is its speed at a single instant?
但它在某一瞬间的速度是多少呢？
To find it, we slide the two points together, closer and closer, until the line just touches the curve at one point.
要求它，我们把这两点慢慢滑到一起，越来越近，直到这条线只在一点上与曲线相切。
That touching line is the tangent, and its slope is the answer.
这条相切的线就是切线，它的斜率就是答案。
That slope is the derivative — the definition of the derivative.
这个斜率就是导数。
Today we define the derivative — the exact rate of change at a single point — and learn the rules that let us find it quickly.
今天我们来定义导数——它是某一点处精确的变化率—— 并学习那些能让我们快速求导的法则。
Let's begin.
让我们开始吧。
Start with the average rate of change over an interval.
先看某个区间上的平均变化率。
Take two points on the curve, and draw the straight line through them.
在曲线上取两点，画出穿过它们的直线。
This line is called a secant.
这条线叫做割线。
Its slope is the change in the output, divided by the change in the input.
它的斜率就是输出的变化，除以输入的变化。
We call this the difference quotient.
我们把它称为差商。
It measures how fast the function changes on average, across the whole interval.
它衡量的是函数在整段区间上平均变化得有多快。
Now the key idea.
现在是关键思想。
Keep one point fixed, and slide the other one toward it.
让一个点固定，把另一个点滑向它。
As the gap shrinks toward zero, the secant line pivots, and settles onto the tangent — the line that just touches the curve.
当间隔缩小到零时， 割线不断转动，最后停在切线上——那条只与曲线相切的线。
Its slope is the instantaneous rate of change: the derivative at that point.
它的斜率就是瞬时变化率：也就是那一点的导数。
In symbols, the derivative is the limit of the difference quotient as the step size goes to zero.
用符号写， 导数就是当步长趋于零时，差商的极限。
When that limit exists, we write it f prime of a.
当这个极限存在时，我们把它记作 f 撇 a。
Let the point vary, and the derivative becomes a whole new function: it gives the slope at every input.
让这个点自由变动，导数就变成一个全新的函数：它在每个输入处给出斜率。
You will see three notations — d y d x, f prime of x, and y prime.
你会看到它有三种写法——d y d x，f 撇 x，以及 y 撇。
They all mean the same thing: the rate of change of the function.
它们的意思完全相同： 都是函数的变化率。
Geometric meaning: the derivative is the slope of the tangent, so we can write the equation of the tangent line.
因为导数是斜率，我们就能写出切线的方程。
It passes through the point on the curve, with a slope of f prime of a.
它经过曲线上那个点， 斜率为 f 撇 a。
Use the point-slope form: y minus f of a, equals f prime of a, times the quantity x minus a.
用点斜式：y 减 f a，等于 f 撇 a，乘以 x 减 a。
Writing this line is a routine exam task, so keep the form ready.
写出这条切线是考试的常规任务，所以要把这个形式记熟。
You will not always have a formula.
你不会总有公式可用。
When a function is given by a table, estimate the derivative with a difference quotient over a small interval around the point.
当函数是用表格给出时，就用该点附近一个小区间上的差商来估计导数。
Use the closest values on each side.
取两侧最接近的值。
And in a real-world model, always attach the units — the output unit, per input unit.
而在实际模型中，一定要带上单位——每一个输入单位对应的输出单位。
The exam asks for this almost every year.
考试几乎每年都会考这一点。
When does a derivative exist?
导数何时存在呢？
Here is the key rule — differentiability implies continuity: if a function is differentiable at a point, then it is continuous there.
关键法则是：如果一个函数在某点可导，那么它在那里一定连续。
But the reverse is not true.
但反过来不成立。
A continuous function can still fail to be differentiable, in two ways.
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 becomes infinite.
在垂直切线处，斜率变成无穷大。
In both cases, there is no single derivative.
这两种情形下，都没有唯一的导数。
From here, we use rules instead of the limit every time.
从现在起，我们用法则，而不是每次都用极限。
The most important is the power rule.
最重要的是幂法则。
To differentiate x to a power, bring the power down to the front, and lower the power by one.
要对 x 的某个幂求导，把幂次拿到前面来，并把幂次减一。
It works for any real power — whole numbers, negative powers, and roots.
它对任何实数幂都成立——整数、负幂、还有根式。
Just rewrite a root or a fraction as a power first, then apply the rule.
只要先把根式或分式写成幂的形式，再套用这条法则。
Three more rules let you differentiate term by term.
还有三条法则让你逐项求导。
The derivative of a constant is zero — a constant never changes.
常数的导数是零——常数从不变化。
A constant multiple just comes along for the ride.
常数倍数只是跟着一起走。
And a sum or a difference is differentiated piece by piece.
而和或差是逐项分别求导的。
Together with the power rule, these differentiate any polynomial, one term at a time.
配合幂法则，这些法则能对任何多项式逐项求导。
These are the derivatives of sin x, cos x, e to the x, and ln x.
这些就是 sin x、cos x、e 的 x 次方和 ln x 的导数。
Four building-block derivatives you must know by heart.
有四个基本导数，你必须背得滚瓜烂熟。
The derivative of sine is cosine.
正弦的导数是余弦。
The derivative of cosine is negative sine — watch that minus sign.
余弦的导数是负的正弦——注意那个负号。
The exponential function is its own derivative; it does not change at all.
指数函数的导数就是它自己，完全不变。
And the derivative of the natural logarithm is one over x.
而自然对数的导数是 x 分之一。
Memorise these four — you will use them everywhere.
把这四个记牢——你会到处用到它们。
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。
What about a product of two functions?
那么两个函数的乘积呢？
You cannot just multiply the two derivatives.
你不能只把两个导数相乘。
Use the product rule: the derivative of the first, times the second, plus the first, times the derivative of the second.
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 example, x squared times e to the x gives two x times e to the x, plus x squared times e to the x.
举个例子， x 平方 乘以 e 的 x 次方，就得到 二 x 乘以 e 的 x 次方，加上 x 平方 乘以 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, all divided by the bottom squared.
对于商，用商法则：分母乘以分子的导数，减去分子乘以分母的导数， 再全部除以分母的平方。
The order matters, because of that minus sign.
次序很重要，因为有那个负号。
This rule even gives the other trig derivatives from identities: write tangent as sine over cosine, apply the rule, and you get one over cosine squared — the secant squared.
这条法则甚至能给出其他三角函数的导数：把正切写成正弦除以余弦，套用法则， 就得到余弦平方分之一——也就是正割的平方。
And here are the derivatives of tan x, cot x, sec x, and csc x.
下面是 tan x、cot x、sec x 和 csc x 的导数。
That same rule gives you the rest of the trigonometric derivatives, and this is the part students try to memorise when they do not have to.
同一条法则能给出其余的三角函数导数， 而这正是学生们明明不必背、却偏要去背的部分。
You are never asked to remember these separately: you rewrite it with an identity, and then use the quotient rule you just learned.
从来没人要求你把它们分别记住：你只要用一个恒等式改写它， 再用刚学的商法则就行。
Tangent is sine over cosine, and the quotient rule turns it into one over cosine squared, which is secant squared.
正切是正弦除以余弦，商法则把它变成一除以余弦的平方，也就是正割的平方。
Cotangent is cosine over sine, and the same work gives minus cosecant squared.
余切是余弦除以正弦，同样的推导给出负的余割平方。
Secant is one over cosine, giving secant times tangent.
正割是一除以余弦，得到正割乘正切。
And cosecant is one over sine, giving minus cosecant times cotangent.
余割是一除以正弦，得到负的余割乘余切。
There is a pattern worth keeping: every co function carries a minus sign — cosine, cotangent and cosecant all differentiate to something negative.
有一个值得记住的规律：每一个带"余"字的函数都带一个负号—— 余弦、余切、余割求导后都是负的。
If you forget one in the exam, rewrite it as a fraction and derive it in two lines.
如果考场上忘了某一个，就把它写成分数，两行推出来。
Let's put it together.
我们把它合起来。
Differentiate x squared, times sine x.
对 x 平方 乘以 正弦 x 求导。
First read the structure: it is a product, so use the product rule.
先看结构： 它是一个乘积，所以用乘积法则。
The first part is x squared, whose derivative is two x.
第一部分是 x 平方，它的导数是 二 x。
The second is sine x, whose derivative is cosine x.
第二部分是 正弦 x，它的导数是 余弦 x。
Now combine: the derivative of the first times the second, plus the first times the derivative of the second.
现在合起来：第一个的导数乘以第二个， 加上第一个乘以第二个的导数。
The answer is two x sine x, plus x squared cosine x.
答案是 二 x 正弦 x，加上 x 平方 余弦 x。
Before you go, three things that save marks.
结束之前，三件能保住分数的事。
First, the definition of the derivative is the slope of the tangent — the limit of the secant slope as the step goes to zero.
第一，导数是切线的斜率—— 当步长趋于零时，割线斜率的极限。
Second, differentiability implies continuity, but never the reverse: a corner is continuous, yet has 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。
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。
Third, read the structure before you differentiate — a product needs the product rule, not the power rule on each factor.
第三，求导之前先看清结构—— 乘积要用乘积法则，而不是对每个因子分别用幂法则。
Get these right, and derivatives are yours.
把这些做对，导数就是你的了。