Course Homepage Preface On Learning
Build Your Intuition
1. 1-Minute Summary 2. X-Ray Vision 3. 3d Intuition
Learn The Lingo
4. Integrals, Derivatives 5. Computer Notation
Basic Understanding
6. Improved Algebra 7. Linear Changes 8. Squared Changes
Deeper Understanding
9. Infinity 10. Derivatives 11. Fundamental Theorem
Figure Out The Rules
12. Add, Multiply, Invert 13. Patterns In The Rules 14. Take Powers, Divide
Put It To Use
15. Archimedes' Formulas Summary
Welcome to an intuition-first calculus course.
Read online, or buy the complete edition for videos, PDF, and discussion.
8 min read

10. The Theory Of Derivatives

The last lesson showed that an infinite sequence of steps could lead to a finite conclusion. Let's put it into practice, and see how breaking change into infinitely small parts can point to the the true amount.

Analogy: Measuring Heart Rates

Imagine you're a doctor trying to measure a patient's heart rate while exercising. You put a guy on a treadmill, strap on the electrodes, and get him running. The machine spit out 180 beats per minute. That must be his heart rate, right?

Nope. That's his heart rate when observed by doctors and covered in electrodes. Wouldn't that scenario be stressful? And what if your Nixon-era electrodes get tangled on themselves, and tug on his legs while running?

Ah. We need the electrodes to get some measurement. But, right afterwards, we need to remove the effect of the electrodes themselves. For example, if we measure 180 bpm, and knew the electrodes added 5 bpm of stress, we'd know the true heart rate was 175.

The key is making the knowingly-flawed measurement, to get a reading, then correcting it as if the instrument wasn't there.

Measuring the Derivative

Measuring the derivative is just like putting electrodes on a function and making it run. For $$$f(x) = x^2$$$, we stick an electrode of $$$+1$$$ onto it, to see how it reacted:

Wolfram Alpha

The horizontal stripe is the result of our change applied along the top of the shape. The vertical stripe is our change moving along the side. And what's the corner?

It's part of the horizontal change interacting with the vertical one! This is an electrode getting tangled in its own wires, a measurement artifact that needs to go.

Throwing Away The Artificial Results

The founders of calculus intuitively recognized which components of change were "artificial" and just threw them away. They saw that the corner piece was the result of our test measurement interacting with itself, and shouldn't be included.

In modern times, we created official theories about how this is done:

There are entire classes dedicated to exploring these theories. The practical upshot is realizing how to take a measurement and throw away the parts we don't need.

Here's the setup, described with limits:

Derivative process

Step Example
Prereq: Start with a function to study $$f(x) = x^2 $$
1: Change the input by dx, our test change $$f(x + dx) = (x + dx)^2 = x^2 + 2x\cdot dx + (dx)^2 $$
2: Find the resulting change in output, $$$df$$$ $$f(x + dx) - f(x) = 2x\cdot dx + (dx)^2 $$
3: Find $$$\frac{df}{dx}$$$ $$\frac{2x\cdot dx + (dx)^2}{dx} = 2x + dx $$
4: Throw away the measurement artifacts $$2x + dx \overset{dx \ = \ 0} \Longrightarrow 2x $$

Wow! We found the official derivative for $$$\frac{d}{dx} x^2$$$ on our own:

derivative of x^2

Now, a few questions:

Practical conclusion: We can start with a knowingly-flawed measurement ($$$f'(x) \sim 2x + dx$$$), and deduce the result it points to ($$$f'(x) = 2x$$$). The theories of exactly how we throw away $$$dx$$$ aren't necessary to master today. The key is realizing there are measurement artifacts that can be removed when modeling how pattern changes.

(Still shaky about exactly how dx can appear and disappear? Good. This question took mathematicians decades to figure out. Here's a deeper discussion of how the theory works, but remember this: When measuring, ignore the effect of the instrument.)

Get Free Lessons And Updates

Join 250k monthly readers and Learn Right, Not Rote.

Class Discussion

Guided class discussions are available in the complete edition.