7.1Finding the Derivative Of A Line

The derivative of a pattern,\( \frac{d}{dx}f(x) \), is the sequence of slices we get as we change an input variable (\( x \)是这里的自然选择)。How do we figure out the sequence of steps?

Well, I imagine going to Home Depot and pestering the clerk:

我想要一些木材。What will it run me?

Clerk: How much do you want?

我想有1英尺高。

Clerk: That’ll be \$4.Anything else I can help you with?

You: Actually, it might be 2 feet.

职员:一共是\ 8美元。Anything else I can help you with?

You: It might be 3 feet.

Clerk:(sigh)这是\ 12美元。Anything else I can help you with?

4英尺怎么样?

We have a relationship (\( f(x) = 4x \)) and investigate it by changing the input a tiny bit.We see if there’s a change in output (there is!), then we change the input again, and so on.

In this case, it’s clear that an additional foot of fencing raises the cost by \$4.So we’ve just determined the derivative to be a constant 4, right?

Not so fast.Sure, we thought about the process and worked it out, but let’s be a little more organized (not every pattern is so simple).我们能描述一下我们的步骤吗?

1. Get the current output,\( f(x) \).In our case,\( f(1) = 4 \).
2. Step forward by\( dx \)(1 foot, for example)
3. Find the new amount,\( f(x + dx) \).In our case, it’s\( f(1 + 1) = f(2) = 8 \).
4. Compute the difference:\( f(x + dx) - f(x) \), or 8 - 4 = 4

Ah!下一步和当前步骤之间的区别是我们切片的大小。For\( f(x) = 4x \)we have:

Increasing length by\( dx \)increases the cost by\( 4 \cdot dx \).

That statement is true, but a little awkward: it talks about the total change.Wouldn’t it be better to have a ratio, such as “cost per foot”?

We can extract the ratio with a few shortcuts:

• \( dx \)= change in our input
• \( df \)= resulting change in our output,\( f(x + dx) - f(x) \)
• \( \frac{df}{dx} \)= ratio of output change to input change

In our case, we have

Notice how we express the derivative as\( \frac{df}{dx} \)instead of\( \frac{d}{dx}f(x) \).到底发生了什么事?It turns out there’s a few different versions we can use.

Think about the various ways we express multiplication:

• Times symbol: 3\( \times \)4 (used in elementary school)
• Dot: 3\( \cdot \)4 (used in middle school)
• Implied multiplication with parentheses:\( (x + 4)(x + 3) \)
• Implied multiplication with a space:\( 2\pi r \ dr \)

The more subtle the symbol, the more we focus on the relationship between the quantities; the more visible the symbol, the more we focus on the computation.

The notation for derivatives is similar:

Some versions, like\( f'(x) \), remind us the sequence of steps is a variation of the original pattern.Notation like\( \frac{df}{dx} \)让我们进入以细节为导向的模式，思考输出变化相对于输入变化的比率(“每增加一英尺的成本是多少?”)。

Remember, the derivative is a complete description of all the steps, but it can be evaluated at a certain point to find the step there: What is the additional cost/foot when\( x = 3 \)?In our case, the answer is 4.

Here’s what the computer returns for this problem:

Nice!As we suspected, the pattern\( f(x) = 4x \)changes by a constant 4 as we increase\( x \).

7.2Finding The Integral Of A Constant

现在让我们从另一个方向研究:给定步骤的顺序，我们能找到原始图案的大小吗?

In our fence-building scenario, it’s fairly straightforward.Solving

means answering “What pattern has an output change of 4 times the input change?”.

Well, we’ve just seen that\( f(x) = 4x \)results in\( f'(x) = 4 \).So, if we’re given\( f'(x) = 4 \), we can guess the original function must have been\( f(x) = 4x \).

I’m pretty sure we’re right (what else could the integral of 4 be?), but let’s compare this with the computer:

Whoa – there’s two different answers (definite and indefinite).Why?Well, there’s many functions that could increase cost by \$4/foot!Here’s a few:

• Cost = \$4 per foot, or\( f(x) = 4x \)
• Cost = \$4 + \$4 per foot, or\( f(x) = 4 + 4x \)
• Cost = \$10 + \$4 per foot, or\( f(x) = 10 + 4x \)

There could be a fixed per-order fee, with the fence cost added in.All the equation\( f'(x) = 4 \)says is that eachadditionalfoot of fencing is \$4, but we don’t know the starting conditions.

• Thedefinite integraltracks the accumulation of a set amount of slices.The range can be numbers, such as\( \int_0^{13} 4 \), which measures the slices from x=0 to x=13 (13\( \cdot \)4 = 52).If the range includes a variable (0 to x), then the accumulation will be an equation (\( 4x \)).
• Theindefinite integralfinds the actual formula that created the pattern of steps, not just the accumulation in that range.It’s written with just an integral sign:\( \int f(x) \).And as we’ve seen, the possibilities for the original function should allow for a starting offset of C.

The notation for integrals can be fast-and-loose, and it’s confusing.Are we looking for an accumulation, or the original function?Are we leaving out\( dx \)?These details are often omitted, so it’s important to feel what’s happening.

7.3The Secret: We Can Work Backwards

The little secret of integrals is that we don’t need to solve them directly.Instead of trying to glue slices together to find out their area, we just learn torecognizethe derivatives of functions we’ve already seen.

If we know the derivative of 4x is 4, then if someone asks for theintegral如果是4，我们可以用“4x”来回应(当然是加上C)。It’s like memorizing the squares of numbers, not the square roots.When someone asks for the square root of 121, dig through and remember that 11\( \times \)11 = 121.

An analogy: Imagine an antiques dealer who knows the original vase just from seeing a pile of shards.

How does he do it?Well, he takes replicas in the back room, drops them, and looks at the pattern of pieces.Then he comes to your pile and says “Oh, I think this must be a Ming Dynasty Vase from the 3rd Emperor.”

他不会试图把你的一堆东西粘在一起——他只是以前见过那个完全一样的花瓶打碎了，而你的一堆东西看起来是一样的!

Now, there may be piles he’s never seen, that aredifficultor impossible to recognize.In that case, the best he can do is to just add up the pieces (with a computer, most likely).He might determine the original vase weighed 13.78 pounds.That’s a data point, fine, but it’s not as nice as knowingwhatthe vase was before it shattered.

This insight was never really explained to me: it’s painful to add up (possibly changing) steps directly, especially when the pattern gets complicated.So, just learn to recognize the pattern from the derivatives we’ve already seen.

7.4Getting To Better Multiplication

把相同大小的步数粘在一起看起来像普通的乘法，对吧?You bet.If we wanted 3 steps (0 to 1, 1 to 2, 2 to 3) of size 2, we might write:

Again, this is a fancy way of saying “Accumulate 3 steps of size 2: what do you get in total?”.我们正在按等量变化的顺序延时。

Now, suppose someone asks you to add 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2.You might say:Geez, can’t you write it more simply?You know, something like:

7.5Creating The Abstract Rules

Have an idea how linear functions behave?Great.We can make a few abstract rules – like working out the rules of algebra for ourselves.

If we know our output is a scaled version of our input (\( f(x) = ax \)), the derivative (pattern of changes) is

and the integral (pattern of accumulation) is

That is, the ratio of each output step to each input step is a constant\( a \)(4, in our examples above).And now that we’ve broken the vase, we can work backwards: if we accumulate steps of size\( a \), they must have come from a pattern similar to\( a \cdot x \)(plus C, of course).

Notice how I wrote\( \int a \)and not\( \int a \ dx \)– I wanted to focus on\( a \), and not details like the width of the step (\( dx \)).Part of calculus is learning to expose the right amount of detail.

One last note: if our output does not reactat allto our input (we’ll charge you a constant \$2 no matter how much you buy… including nothing!)then “steps” are a constant 0:

In other words, there is no difference in the before-and-after measurement.Now, a pattern may have anoccasionalzero slice, if it stands still for a moment.That’s fine.But ifevery切片为零，意味着我们的模式从未改变。

有一些微妙的方法，但让我们学会在“真的，我饿了”之前说“我想要食物”。