13.1Combining Perspectives

Imagine a business with interacting departments, or a machine with interconnected parts.What happens when we make a change?There’s a potential impact on each part.

如果我们的业务有4个部门，我们做一个政策改变，有4个角度需要考虑。It sounds simple when written out: whenever a change happens, see what happens to each part!

No matter the specific interaction between parts F and G (addition, subtraction, multiplication, exponents…), we just have two perspectives to consider:

Aha!That’s why the rules for\( (f + g)' \)and\( (f \cdot g)' \)are two added perspectives.The derivative of\( \frac{1}{X} \)has a perspective because there’s just one “moving part”, x.(If you like, thereisa contribution from “1” about the change it experiences: nothing.不管你吼多少声，1还是1。)

The exact contribution from a perspective depends on the interaction:

• With addition, each part adds a direct change (\( df + dg \)).
• With multiplication, each part thinks it’ll add a rectangular strip (\( f \cdot dg + g \cdot df \)).(I’m using\( df \)instead of\( f' \)to help us think about the slice being added.)

You might forget the exact form of the multiplication rule.But you can think “The derivative of\( f \cdot g \)must be something with\( df \)+ something with\( dg \).”

Let’s go further: what about the derivative of\( a \cdot b + c \)?You guessed it, 3 perspectives that should be added: something involving\( da \)plus something involving\( db \)plus something involving\( dc \).

We can predict the shape of derivatives for gnarly equations.What’s the derivative of:

Wow.I can’t rattle that off, but I can say it’ll be something involving 4 additions (\( dx \),\( dy \),\( du \)and\( dv \)).Guess the shape of a derivative, even if you don’t know the exact description.

Why does this work?Well, suppose we had a change that was influenced by both\( da \)and\( db \), such as\( 15 \cdot da \cdot db \)– that’d be our instrument interfering with itself!

Only direct changes on a single variable’s are counted (such as\( 3 da \)or\( 12 db \)), and “changes on changes’’ like\( 15 \cdot da \cdot db \)are ignored.

13.2Dimensional Intuition

Remember that derivatives are a fancier form of division.What happens when we do a division like\( \frac{x^3}{x} \)?We divide volume by length, and get area (one dimension down).

What happens when we do\( \frac{d}{dx} x^3 \)?You might not know yet, but you can bet we’ll be dropping a dimension.

When we divide or take a derivative, we drop a dimension and hop down the table.Volume is built from area (slices\( dx \)thick), area is built from length (\( dx \)wide).

A constant value, like 3, has no dimension in the following sense: it never produces a set of slices.永远不会跳转到下一个值。Once we have a constant value, we get “stuck” on the table because the 0 pattern always produces another 0 pattern (volume to area to lines to constant to zero to zero…).

We can have negative dimensions as well: “per length”, “per area”, “per volume”, etc.Derivatives still decrease the dimension, so when seeing\( \frac{1}{x} \), we know the derivative will resemble\( \frac{1}{x^2} \)当维度下降一个层次。

An important caveat: Calculus only cares about quantities, not their dimension.The equations will happily combine\( x \)and\( x^2 \)，尽管我们知道长度和面积不能混在一起。Units add a level of meaning fromoutsidethe equation, that help keep things organized and warn us if we’ve gone awry (if you differentiate area and get volume, you know something went wrong).

13.3Thinking With Dimensions

你可以考虑导数的维数而不用深入研究特定的公式。Imagine the following scenario:

• Take a string and wrap it tight around a quarter.Take another string and wrap it tight around the Earth.
• Lengthen both strings, adding more to the end, so there’s a 1-inch gap all the way around the quarter, and a 1-inch gap all the way around the Earth (like having a ring floating around Saturn).
• Quiz: Which scenario needed more extra string to create?Is it more string to put a 1-inch gap around the quarter, or a 1-inch gap around the Earth?

We can crunch through the formula, but think higher-level.Because circumference is a 1-d line, its reaction to change (the derivative) will be a constant.No matter the current size, the circumference will change the same amount with every push.So, the extra string needed is the same for both!(About 6.28 inches per inch of radius increase).

Now, suppose we were painting the sphere instead of putting a string around it.Ah, well, area is squared, therefore the derivative is a dimension lower (linear).If we have a 5-inch and 10-inch sphere, and make them 1-inch bigger each, the larger sphere will require double theextrapaint.

13.4Questions

Let’s think about the derivative of\( x^3 \), a growing cube.

1) What dimension should the derivative of\( x^3 \)have?

2) How many viewpoints should\( x^3 = x \cdot x \cdot x \)involve?

3) Have a guess for the derivative?Does it match with how you’d imagine a cube to grow?