Exponential and Log

MIT 18.01 Single Variable Calculus, lecture 6, explained in plain English

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Part 1 · 0:00–10:00

This part introduces exponential functions like 2^x, sets up their basic rules, and starts the work of finding their derivative.

Key ideas

A base is a positive number, like 2 in 2^x. Any base raised to the power 0 equals 1, and a^1 equals a. 1:04
The basic rule of exponents says a^(x1 + x2) equals a^x1 times a^x2. Together with the base rules, this rule defines the exponential function. 1:26
The professor says the graph of 2^x grows very fast. Points like (0, 1), (-1, 1/2), and (1, 2) help you picture it. 4:08
To find the derivative of a^x, you plug a^x into the definition of the derivative. The rule a^(x + delta x) = a^x times a^(delta x) lets you pull out a common factor a^x. 6:29
When taking the limit, delta x is the moving variable and x stays fixed. That means a^x is a constant and can be pulled out of the limit. The derivative then becomes a^x times a mystery number M(a). 7:15

Words to know

base: The number that gets raised to a power. In a^x, the base is a.
exponential function: A function where the variable is in the power, like y = 2^x, instead of the base.
derivative: A measure of how fast a function changes at a point. It is found using a limit.
limit: The value a expression gets closer and closer to as a small quantity, like delta x, gets closer to zero.

Example

Think of 2^x as doubling. Each time x goes up by 1, the value doubles, so 2^0 is 1, 2^1 is 2, and 2^2 is 4. This is why exponential graphs shoot up so quickly.

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

Part 2 · 10:00–20:00

This part explains why the special number e exists and why the slope of e^x at any point equals e^x itself.

Key ideas

M(a) is the slope of the graph of a^x at x = 0, the point where the tangent line touches the curve. Finding M(a) is the main problem the lecture is working on. 10:19
If you know the slope at one single point, you can work out the slope everywhere. The lecturer says this is the same method used earlier for sine and cosine. 11:00
Instead of finding M(a) directly, the lecturer defines a number e as the one whose M(e) equals 1. We don't know its value yet, but we assume it exists and check later. 13:14
With this definition, the derivative of e^x is just e^x. The lecturer calls this the most important formula from the lesson. 14:06
Stretching a function by a factor k multiplies its slope at 0 by k. So by choosing the right k, you can make the slope exactly 1, which shows that e exists. 17:51

Words to know

derivative: A measure of how fast a function is changing at a point, found by a formula.
slope: How steep a line or curve is at a point. A bigger slope means steeper.
tangent line: A straight line that just touches a curve at one point and has the same steepness as the curve there.
chain rule: A rule for finding the derivative of a function that is inside another function, such as f(kx).

Example

Think of a hill with a steep spot. If you stretch the hill sideways, the same rise happens over a longer distance, so it gets less steep, and if you squeeze it, it gets steeper. Multiplying the stretch by a number changes the steepness by that same number, which is the idea the lecturer uses to pick the right base.

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

Part 3 · 20:00–30:00

This part explains the natural log, how to find its rate of change, and how to convert any exponential to base e so it can be differentiated.

Key ideas

The natural log, written ln(x), is the inverse of e^x. If y = e^x, then the log of y gives back x. 20:40
The log turns multiplication into addition: ln(x1 times x2) equals ln(x1) plus ln(x2). Also, ln(1) is 0 and ln(e) is 1. 21:58
The graph of ln x is the mirror image of e^x across the diagonal line x = y. It crosses the horizontal axis at (1, 0) and only exists for positive x. 23:28
The rate of change of ln x is 1/x. This comes from implicit differentiation: write w = ln x, rewrite it as e^w = x, then differentiate both sides. 25:22
To differentiate any exponential a^x, rewrite it in base e as e^(x ln a). Then you can use the rules for e^x. 29:20

Words to know

inverse function: A function that undoes another function. If one function takes x to y, its inverse takes y back to x.
implicit differentiation: A way to find a rate of change when the equation is not written in the form y = something. You take the rate of change of both sides, even when the variables are mixed together.
chain rule: A rule for finding the rate of change of a function that is inside another function. You multiply the rate of change of the outer part by the rate of change of the inner part.
natural log: The logarithm with base e, written ln(x). It answers the question: what power of e gives x?

Example

Think of an undo button on a computer. Pressing undo reverses the last action and brings you back to where you started. The inverse function works the same way: if e^x takes you from x to y, the natural log takes you back from y to x.

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

Part 4 · 30:00–40:00

This part shows how to find the rate of change of a^x, first by using the chain rule and then by a method called logarithmic differentiation, and it finds that the natural log is the key number.

Key ideas

The rate of change of e^(3x) follows the chain rule: you multiply by the rate of change of the exponent, which is 3. The same idea works for a^x. 30:23
The number ln a looks scary, but it is just a fixed number that does not move with x. Treat it as a constant that scales the answer. 30:43
The derivative of a^x is (ln a) times a^x. The number ln a is the natural log of a, and it appears even when you use base 2 or base 10. 32:11
Logarithmic differentiation takes the log of the function first. The rule (ln u)' = u'/u turns a hard problem into an easier one, and then you multiply back by u at the end. 35:31
The lecturer says both methods always work, and the second is useful for harder functions where both the base and the exponent change, such as x^x. 39:12

Words to know

derivative: How fast a function's output changes when its input changes. It is the rate of change.
chain rule: A rule for finding the rate of change of a function inside another function. You find the outer change and multiply by the inner change.
natural logarithm (ln): A special log, written ln, that uses the number e as its base. It shows up naturally in many calculus results.
constant: A number that does not change. In this lesson, ln a is a constant because a does not change as x changes.

Example

Think of a bank account that grows by a fixed percent each year. The growth speed depends on the current amount, and the fixed percent is like ln a: it does not change, but it sets how fast things grow. Taking logs first turns the growth into a simple multiplication, which is easier to track.

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

Part 5 · 40:00–47:56

This part shows how to take the derivative of x to the power of x using the product rule, then finds that the limit of (1 + 1/n) to the power n is exactly e by using logarithms.

Key ideas

When the exponent is a variable, you can use the product rule after taking the natural log of both sides. This gives the derivative of x^x. 40:00
The result is that the derivative of x^x equals x^x times (1 + ln x). The x^x part is kept, and the other factor comes from the product rule. 40:45
To find the limit of (1 + 1/n)^n, take the natural log first. This turns the changing exponent into a simpler product, n times ln(1 + 1/n). 42:36
Let delta x equal 1/n. As n goes to infinity, delta x goes to 0, and the problem becomes a difference quotient for the derivative of ln x at x = 1. That derivative is 1, so the log of the limit is 1. 44:18
Undo the log with the exponential, so the limit is e to the power 1, which equals e. This gives a practical way to calculate e, such as using 1 + 1/100 to the power 100. 46:56

Words to know

product rule: A rule for finding the derivative of two factors multiplied together. You take the derivative of one factor, keep the other, then do it the other way around and add the results.
limit: The value that a expression gets closer and closer to as a number gets closer to some point, such as n going to infinity.
difference quotient: A fraction that compares how much a function changes to how much its input changes. Its limit gives the derivative.
reciprocal: The number you get when you flip a fraction. For example, the reciprocal of n is 1/n.

Example

Think of a moving exponent like a staircase that keeps getting taller. Taking the log is like writing the height as a simple list of steps you can count, so the hard problem becomes an easier one. Once you know the simple answer, you can convert it back to the original form.

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

← Lecture 4: Chain Rule · Lecture 16: Differential Equations →

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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