Derivatives

MIT 18.01 Single Variable Calculus, lecture 3, explained in plain English

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Part 1 · 0:00–10:00

This part explains two kinds of derivative formulas and starts the work of finding the derivatives of sine and cosine.

Key ideas

Derivative formulas come in two kinds. Specific formulas give the derivative of one particular function, such as x^n. General formulas tell you how derivatives behave when you add functions or multiply by a constant. 0:25
To find the derivative of sine, you start with the difference quotient: sin(x + delta x) minus sin x, divided by delta x. Then you take the limit as delta x goes to 0. The sum formula for sine is used to rewrite the top part. 3:15
When delta x is set to 0, the expression becomes 0 divided by 0, which is tricky. You must group the terms so that a 0 stays over a 0. If a 1 over 0 appears, the answer has no meaning. 5:52
Two limit facts are used. As delta x goes to 0, (cos delta x minus 1) divided by delta x goes to 0, and sin(delta x) divided by delta x goes to 1. The lecturer says these will be checked later. 7:25
Putting it together, the derivative of sine is cosine. The calculation for cosine starts the same way, but it uses the sum formula for cosine. 8:34

Words to know

derivative: A measure of how fast a function's output changes as its input changes, found using a limit.
difference quotient: A fraction that compares how much a function changes over a small step, divided by that step size.
limit: The value a expression gets closer and closer to as a small number, like delta x, gets closer to 0.
constant: A fixed number that does not change, such as c in the formula c times u.

Example

Think of a car's speedometer. It shows how fast you are going right now, not over a long trip. A derivative works the same way: it finds the rate of change at one point by looking at very small steps, like checking the car over a tiny slice of time.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part shows why the rates of change of sine and cosine work out the way they do, and gives a picture-based proof that sin θ / θ gets close to 1 for small angles.

Key ideas

The derivative of cos x works out to negative sin x. The lesson gets there by regrouping the terms and using the two properties called A and B. 10:59
To find the rates of change of sine and cosine everywhere, you only need their rates of change at one point, x = 0. The lesson uses that point to find the rest. 12:30
At x = 0, the rate of change of cos x comes out to the limit that property A says is 0. The rate of change of sin x at x = 0 comes out to the limit that property B says is 1. 12:51
Property B has to be proved with geometry, because sine and cosine are described geometrically here. The proof uses a unit circle, a picture of an angle, and a doubled 'bow' shape. 15:29
The ratio sin θ / θ gets close to 1 for small angles, because a very short curved arc looks almost straight, so the two lengths almost match. 19:11

Words to know

limit: The value that an expression gets closer and closer to as a small number, here delta x, gets closer and closer to zero.
derivative: How fast a function's output changes when its input changes. The lesson uses it to describe the rate of change of sine and cosine.
unit circle: A circle with a radius of exactly 1, centered on the origin, which makes it easy to draw sine and cosine as distances.
radians: A way of measuring angles using arc length. The lesson says an angle theta also gives an arc of that length when measured in radians.

Example

The lesson's bow picture shows a bent bow and its string. A short arc of a circle looks almost like a straight line, so the bent part and the string are nearly the same length. The same idea explains why sin θ / θ gets close to 1 when θ is very small.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part explains why a tiny piece of a curve looks almost straight, and why angles must be measured in radians for the math to work.

Key ideas

Short pieces of curves are nearly straight. If you zoom in far enough on a curve, it looks like a straight line. 20:00
The professor proves the key fact by picturing a circle with a vertex (the point where the two sides of an angle meet) that is pushed far away. The curved arc then looks more and more like a straight vertical line. 22:07
The gap between the curve and the straight line is 1 minus cos theta. As theta gets smaller, this gap gets smaller compared with the length it is measured against, so the ratio goes to 0. 23:09
The denominator (theta) also goes to 0, but the gap shrinks much faster than theta does. So the ratio still settles to a clear value. 26:08
The formulas only work when angles are measured in radians. Radians measure an angle as a length along the unit circle, so the arc and the straight line can be compared fairly. Degrees would give wrong answers. 28:32

Words to know

radian: A way to measure an angle using length. It is the length of the arc along a circle of radius 1 that the angle cuts off.
arc length: The length of a curved piece of a circle, measured along the curve itself.
vertex: The corner point where the two lines of an angle meet.
cosine (cos theta): In this picture, the distance from the vertex straight across to the vertical line, when the angle is theta and the main line is length 1.

Example

Think of a tiny piece of a basketball's surface right under your finger. It feels flat, even though the whole ball is round. That is like zooming in on a curve: the small piece looks almost straight.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

The teacher uses pictures of a very small angle and a circle to show how a sine value changes, and why a curved arc can be treated as a straight line for small pieces.

Key ideas

Picture the angle theta as a bird's beak slowly closing. As theta gets smaller and smaller, the picture has to be blown up so you can still see it. 31:25
Exactly at theta = 0 the picture is meaningless, just a flat line. The useful information comes from being just a little away from zero. 32:09
The lesson says a factor of 2 does not matter when you check whether something goes to 0. What matters is how the sizes shrink. 33:19
The second proof treats sine as the height of a point moving around a unit circle. The change in height is the vertical distance PR when the point moves a small angle. 36:22
A tiny piece of a curved arc is replaced with a straight line. That turns the problem into a right triangle, which is much easier to work with. 39:16

Words to know

limit: What a value gets closer and closer to as something else gets closer and closer to a certain number, without necessarily reaching it.
unit circle: A circle with a radius of exactly 1. The lesson defines its radius to be 1 on purpose.
approximation: A simpler stand-in for something complicated that is close enough for the job. Here, a straight line stands in for a small curved arc.
arc: A piece of the edge of a circle, like a curved section of a round ring.

Example

Think of looking closely at a small piece of a big round wheel. The tiny piece looks almost like a straight line, even though the whole wheel is curved. This is why the lesson can swap a small curved arc for a straight line.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–49:53

This part shows, using a circle and small straight pieces, why the derivative of sine is cosine, and then previews the product rule and quotient rule.

Key ideas

A short piece of a curve is almost a straight line. That idea lets us swap a tiny curved arc for a straight segment and do simple geometry. 40:00
For a tiny angle change delta theta, the arc length is about delta theta. So the straight segment PQ is also about delta theta long. 40:24
The line PQ is almost perpendicular to the radius OP. Rotating the picture by 90 degrees shows that the angle at P is about theta. 42:35
Using the triangle with hypotenuse delta theta and angle theta, the vertical side is about delta theta times cos theta. Dividing by delta theta and taking the limit gives the derivative of sine as cosine. 45:48
Two general rules are introduced: the product rule for multiplying two functions, and the quotient rule for dividing them. The lecturer says the proofs come later. 47:50

Words to know

derivative: A measure of how fast something changes. It tells you the slope of a curve at a point.
limit: The value something gets closer and closer to as a small amount shrinks toward zero.
perpendicular: Meeting at a right angle, like the corner of a square.
hypotenuse: The longest side of a right triangle, the one across from the right angle.

Example

Think of a bicycle wheel rolling along the ground. A tiny bit of the rim looks almost straight, just like a small piece of a big circle looks like a line when you zoom in very close. Zooming in on any curve makes it look straight, which is the idea this lesson uses.

3 checked practice questions for this part are in the app.

← Lecture 1: Rate of Change · Lecture 4: Chain Rule →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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