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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 AP-style problems
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

  • Rate of change: Compare average and instantaneous rates of change.
  • Limit definitions: Find a derivative function using the formal limit definitions.

Average rate of change

The average rate of change (ARC) of a function f(x) over the interval [a,b] calculates the change in output per unit input. To calculate it, use the formula:

Average rate of change=b−af(b)−f(a)​

Geometrically, this 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 on [−3,−1].

(spoiler)

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

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

Instantaneous rate of change

While the average rate of change describes behavior over an interval, many quantities change unevenly.

Instead of measuring an average over a span of time, the instantaneous rate of change (IROC) tells us how fast a variable is changing right now, at one exact instant. To describe the rate of change at a single moment, we analyze what happens as the interval shrinks toward zero.

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 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 find the derivative. 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.

Solution

(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​

A different approach: We could also find the general expression for f′(a) first by keeping a as a variable:

(spoiler)

f′(a)​=x→alim​x−ax2−a2​=x→alim​x−a(x−a)(x+a)​=x→alim​(x+a)=2a​

Plugging in a=−3 gives f′(−3)=2(−3)=−6.

Alternative limit definition

The first definition defines the derivative at a point a, although a can be kept as a variable during the limit process to give a general expression like f′(a)=2a. However, to find a derivative function f′(x) using standard variable notation, a second definition is more commonly used. Instead of moving x toward a, this form shrinks the actual distance, h, between two points.

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 obtain 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 will lead to the same answer for any problem, 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

  • Rate of change: Compare average and instantaneous rates of change.
  • Limit definitions: Find a derivative function using the formal limit definitions.

Average rate of change

The average rate of change (ARC) of a function f(x) over the interval [a,b] calculates the change in output per unit input. To calculate it, use the formula:

Average rate of change=b−af(b)−f(a)​

Geometrically, this 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 on [−3,−1].

(spoiler)

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

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

Instantaneous rate of change

While the average rate of change describes behavior over an interval, many quantities change unevenly.

Instead of measuring an average over a span of time, the instantaneous rate of change (IROC) tells us how fast a variable is changing right now, at one exact instant. To describe the rate of change at a single moment, we analyze what happens as the interval shrinks toward zero.

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 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 find the derivative. 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.

Solution

(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​

A different approach: We could also find the general expression for f′(a) first by keeping a as a variable:

(spoiler)

f′(a)​=x→alim​x−ax2−a2​=x→alim​x−a(x−a)(x+a)​=x→alim​(x+a)=2a​

Plugging in a=−3 gives f′(−3)=2(−3)=−6.

Alternative limit definition

The first definition defines the derivative at a point a, although a can be kept as a variable during the limit process to give a general expression like f′(a)=2a. However, to find a derivative function f′(x) using standard variable notation, a second definition is more commonly used. Instead of moving x toward a, this form shrinks the actual distance, h, between two points.

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 obtain 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 will lead to the same answer for any problem, 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

  • AP-style problems