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Introduction
1. Limits
2. Derivative basics
2.1 The derivative
2.1.1 Limit definition of the derivative
2.1.2 Additional problem types
2.2 Tangent lines & slopes
2.3 Power rule
2.4 Product & quotient rules
2.5 Special derivatives
2.6 Differentiability & continuity
3. Advanced differentiation
4. Contextual uses
5. Analytical uses
6. Integration
7. Differential equations
8. Applications of integrals
Wrapping up
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2.1.1 Limit definition of the derivative
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2. Derivative basics
2.1. The derivative
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Limit definition of the derivative

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What you’ll learn

  • Average vs. instantaneous rate of change
  • Finding a derivative with the limit process

Average rate of change

To describe how fast something is changing, we use a rate of change. For example, think about your phone’s battery percentage throughout the day. If you check it at two different times, time a and time b, then the average rate of change is the change in battery percentage divided by the time that passed:

Average rate of change=b−af(b)−f(a)​

This tells you how quickly the battery drained on average between time a and time b. With that information, you can estimate how long the battery might last before needing a recharge (a simplified version of what your phone does).

The average rate of change is simply the slope of a line, from algebra: the change in y-values divided by the change in x-values for the line connecting two points on f(x).

Figure 2.1.1 Average rate of change
Figure 2.1.1 Average rate of change

Example

Find the average rate of change of f(x)=x2 over the interval [−3,−1].

(spoiler)

Using the formula, the average rate of change of f over [−3,−1] is the slope

b−af(b)−f(a)​=−1−(−3)f(−1)−f(−3)​=21−9​=−4

Instantaneous rate of change

In real life, the battery doesn’t deplete at a steady rate. Watching videos uses more power, while leaving the phone idle slows the drain. So how do we describe how quickly the battery is draining at one specific moment?

The average rate of change gives a general picture, but the instantaneous rate of change focuses on what’s happening right at a particular time. The idea is to “zoom in” on a very small time interval.

This is where limits come in. Instead of using two fixed points a and b, we:

  • Fix the point a.
  • Let another point x move closer and closer to a.
  • Track what happens to the average rate of change as x→a.

As x approaches a, the time interval shrinks toward zero, and the limiting value gives the instantaneous rate of change at a. This is also called the derivative of f(x) at x=a:

Definition of the derivative:

f′(a)=x→alim​x−af(x)−f(a)​

The notation f′(a) is read as “f prime of a.” The prime symbol (′) is commonly used to indicate a derivative. Later, you’ll also see second derivatives, written as f′′(a) (“f double prime of a”).

Figure 2.1.2 Instantaneous ROC form 1
Figure 2.1.2 Instantaneous ROC form 1
AP tip:

When asked for the average rate of change, use the basic slope formula with two points - no limits needed.

If it’s the instantaneous rate of change, you must use the derivative or the limit definition. Watch out for multiple-choice traps that mix these up!

Example

Find f′(−3), the instantaneous rate of change of f(x) at x=−3, for the function f(x)=x2.

(spoiler)

Using the limit definition of the derivative with a=−3,

f′(−3)​=x→−3lim​x−(−3)f(x)−f(−3)​=x→−3lim​x+3x2−9​=x→−3lim​(x+3)​(x−3)(x+3)​​=x→−3lim​(x−3)=−6​

Alternative limit definition

The first definition gives the derivative at a specific point a. We can also define a derivative function f′(x), which gives the instantaneous rate of change at any input value x.

This time:

  1. Fix a point x.
  2. Move a small amount h away from it.

In this form, the average rate of change is called the difference quotient:

hf(x+h)−f(x)​

which is the slope between the two points shown in the image below.

Figure 2.1.3 Difference quotient
Figure 2.1.3 Difference quotient

To get the instantaneous rate of change at x, we make the interval as small as possible by letting h→0, bringing (x+h) to x. This results in the alternative limit definition of the derivative:

Alternative definition of the derivative:

f′(x)=h→0lim​hf(x+h)−f(x)​

This limit process creates a new function f′(x). Plugging in a value of x gives the instantaneous rate of change at that point.

Both limit definitions can be used, but one may be easier to work with than the other depending on the function.

Examples

  1. Find the derivative of f(x)=x2.
(spoiler)

Since f(x)=x2, then f(x+h)=(x+h)2.

Then by the limit definition,

f′(x)f′(x)​=h→0lim​hf(x+h)−f(x)​=h→0lim​h(x+h)2−x2​=h→0lim​hx2+2xh+h2−x2​​=h→0lim​h2xh+h2​=h→0lim​h​h​(2x+h)​=h→0lim​(2x+h)=2x​

Notice that inputting x=−3 gives f′(3)=−6, the same result found using the 1st form.

  1. Find f′(x) for the function defined by f(x)=x​, and use the derivative to find the instantaneous rate of change of f at x=9.
(spoiler)

Since f(x)=x​, then f(x+h)=x+h​.

Then by the limit definition,

f′(x)​=h→0lim​hf(x+h)−f(x)​=h→0lim​hx+h​−x​​​

Rationalizing using the conjugate:

=====​ h→0lim​hx+h​−x​​⋅x+h​+x​x+h​+x​​ h→0lim​h(x+h​+x​)(x+h)−x​ h→0lim​h​(x+h​+x​)h​​ h→0lim​x+h​+x​1​ x+0​+x​1​ 2x​1​​

Then evaluating at x=9,

f′(9)=29​1​=61​

Average rate of change

  • Slope between two points on f(x): b−af(b)−f(a)​
  • No limits needed — basic algebra formula

Instantaneous rate of change (derivative at a point)

  • Rate of change at one specific point, found via limit
  • Definition: f′(a)=limx→a​x−af(x)−f(a)​
  • Notation: f′(a) = “f prime of a”; f′′(a) = second derivative

Alternative (limit) definition of the derivative

  • Produces derivative function f’(x) valid for any x
  • Uses difference quotient: f′(x)=limh→0​hf(x+h)−f(x)​
  • Cancel h algebraically (factor or rationalize) before evaluating limit

Key distinctions

  • Average ROC → use slope formula with two points
  • Instantaneous ROC → must use derivative/limit definition
  • Both limit forms yield the same result; choose whichever is easier for the given function

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Limit definition of the derivative

What you’ll learn

  • Average vs. instantaneous rate of change
  • Finding a derivative with the limit process

Average rate of change

To describe how fast something is changing, we use a rate of change. For example, think about your phone’s battery percentage throughout the day. If you check it at two different times, time a and time b, then the average rate of change is the change in battery percentage divided by the time that passed:

Average rate of change=b−af(b)−f(a)​

This tells you how quickly the battery drained on average between time a and time b. With that information, you can estimate how long the battery might last before needing a recharge (a simplified version of what your phone does).

The average rate of change is simply the slope of a line, from algebra: the change in y-values divided by the change in x-values for the line connecting two points on f(x).

Example

Find the average rate of change of f(x)=x2 over the interval [−3,−1].

(spoiler)

Using the formula, the average rate of change of f over [−3,−1] is the slope

b−af(b)−f(a)​=−1−(−3)f(−1)−f(−3)​=21−9​=−4

Instantaneous rate of change

In real life, the battery doesn’t deplete at a steady rate. Watching videos uses more power, while leaving the phone idle slows the drain. So how do we describe how quickly the battery is draining at one specific moment?

The average rate of change gives a general picture, but the instantaneous rate of change focuses on what’s happening right at a particular time. The idea is to “zoom in” on a very small time interval.

This is where limits come in. Instead of using two fixed points a and b, we:

  • Fix the point a.
  • Let another point x move closer and closer to a.
  • Track what happens to the average rate of change as x→a.

As x approaches a, the time interval shrinks toward zero, and the limiting value gives the instantaneous rate of change at a. This is also called the derivative of f(x) at x=a:

Definition of the derivative:

f′(a)=x→alim​x−af(x)−f(a)​

The notation f′(a) is read as “f prime of a.” The prime symbol (′) is commonly used to indicate a derivative. Later, you’ll also see second derivatives, written as f′′(a) (“f double prime of a”).

AP tip:

When asked for the average rate of change, use the basic slope formula with two points - no limits needed.

If it’s the instantaneous rate of change, you must use the derivative or the limit definition. Watch out for multiple-choice traps that mix these up!

Example

Find f′(−3), the instantaneous rate of change of f(x) at x=−3, for the function f(x)=x2.

(spoiler)

Using the limit definition of the derivative with a=−3,

f′(−3)​=x→−3lim​x−(−3)f(x)−f(−3)​=x→−3lim​x+3x2−9​=x→−3lim​(x+3)​(x−3)(x+3)​​=x→−3lim​(x−3)=−6​

Alternative limit definition

The first definition gives the derivative at a specific point a. We can also define a derivative function f′(x), which gives the instantaneous rate of change at any input value x.

This time:

  1. Fix a point x.
  2. Move a small amount h away from it.

In this form, the average rate of change is called the difference quotient:

hf(x+h)−f(x)​

which is the slope between the two points shown in the image below.

To get the instantaneous rate of change at x, we make the interval as small as possible by letting h→0, bringing (x+h) to x. This results in the alternative limit definition of the derivative:

Alternative definition of the derivative:

f′(x)=h→0lim​hf(x+h)−f(x)​

This limit process creates a new function f′(x). Plugging in a value of x gives the instantaneous rate of change at that point.

Both limit definitions can be used, but one may be easier to work with than the other depending on the function.

Examples

  1. Find the derivative of f(x)=x2.
(spoiler)

Since f(x)=x2, then f(x+h)=(x+h)2.

Then by the limit definition,

f′(x)f′(x)​=h→0lim​hf(x+h)−f(x)​=h→0lim​h(x+h)2−x2​=h→0lim​hx2+2xh+h2−x2​​=h→0lim​h2xh+h2​=h→0lim​h​h​(2x+h)​=h→0lim​(2x+h)=2x​

Notice that inputting x=−3 gives f′(3)=−6, the same result found using the 1st form.

  1. Find f′(x) for the function defined by f(x)=x​, and use the derivative to find the instantaneous rate of change of f at x=9.
(spoiler)

Since f(x)=x​, then f(x+h)=x+h​.

Then by the limit definition,

f′(x)​=h→0lim​hf(x+h)−f(x)​=h→0lim​hx+h​−x​​​

Rationalizing using the conjugate:

=====​ h→0lim​hx+h​−x​​⋅x+h​+x​x+h​+x​​ h→0lim​h(x+h​+x​)(x+h)−x​ h→0lim​h​(x+h​+x​)h​​ h→0lim​x+h​+x​1​ x+0​+x​1​ 2x​1​​

Then evaluating at x=9,

f′(9)=29​1​=61​

Key points

Average rate of change

  • Slope between two points on f(x): b−af(b)−f(a)​
  • No limits needed — basic algebra formula

Instantaneous rate of change (derivative at a point)

  • Rate of change at one specific point, found via limit
  • Definition: f′(a)=limx→a​x−af(x)−f(a)​
  • Notation: f′(a) = “f prime of a”; f′′(a) = second derivative

Alternative (limit) definition of the derivative

  • Produces derivative function f’(x) valid for any x
  • Uses difference quotient: f′(x)=limh→0​hf(x+h)−f(x)​
  • Cancel h algebraically (factor or rationalize) before evaluating limit

Key distinctions

  • Average ROC → use slope formula with two points
  • Instantaneous ROC → must use derivative/limit definition
  • Both limit forms yield the same result; choose whichever is easier for the given function

More from The derivative

  • Additional problem types