Topic 2.2 Notes – Defining the Derivative of a Function and Using Derivative Notation

1. The Derivative as a Limit

You already know the average rate of change on an interval $[x, x+h]$:

$$\frac{f(x+h)-f(x)}{h}$$

That fraction is called the difference quotient. Geometrically, it’s the slope of a secant line through two points on the graph.

To get the instantaneous rate of change, we shrink the interval:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$

This limit, if it exists, defines the derivative.

Key ideas to lock in:

• The derivative is a function.
• $f'(x)$ gives the slope of the tangent line at each $x$.
• If the limit does not exist, the function is not differentiable at that point.

Here’s the geometry behind it. The graph below shows a curve (like $y = x^2$) with a secant line through two points and the tangent line at one of them.

Secant lines approaching a tangent line as $h \to 0$

As $h \to 0$, the secant line becomes the tangent line. That visual connection is the entire reason the limit definition matters.

2. Derivative Notation

If $y = f(x)$, all of these mean the same thing:

• $f'(x)$
• $y'$
• $\dfrac{dy}{dx}$

They just emphasize different viewpoints:

• $f'(x)$: derivative as a function
• $f'(a)$: a number (the slope at $x=a$)
• $\dfrac{dy}{dx}$: rate language, “change in $y$ with respect to $x$”

On quizzes and the AP exam, notation switches constantly. If a problem says “find $\frac{dy}{dx}$,” it’s not a different task. It’s the derivative.

3. Using the Limit Definition to Find the Derivative

If a problem says “use the definition,” you must start with the limit. No shortcuts.

The Process

1. Write

   $$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$

2. Carefully compute $f(x+h)$. This is where most mistakes happen.
3. Expand completely.
4. Combine like terms.
5. Factor out $h$ from the numerator.
6. Cancel the $h$.
7. Take the limit by plugging in $h=0$.

If the $h$ doesn’t cancel, check your algebra.

Quick Example

Let $f(x)=2x^2-5x$.

• $f(x+h)=2(x+h)^2-5(x+h)$
• Expand:

  $$2(x^2+2xh+h^2)-5x-5h$$

• Subtract $f(x)$, simplify:

  $$4xh+2h^2-5h$$

• Factor:

  $$h(4x+2h-5)$$

• Cancel $h$, take the limit:

$$f'(x)=4x-5$$

This process shows up often on no-calculator multiple choice because it tests algebra control.

4. Tangent Lines

The derivative at a point gives the slope of the tangent line:

$$f'(a)=\text{slope at } x=a$$

To find the equation of the tangent line at $x=a$:

1. Find $f'(x)$.
2. Compute $m=f'(a)$.
3. Use point-slope form:

   $$y-f(a)=m(x-a)$$

Example:
If $f(x)=x^3$ and you want the tangent line at $x=2$:

• $f'(x)=3x^2$
• $f'(2)=12$
• Point is $(2,8)$

Tangent line:
$$y-8=12(x-2)$$

On FRQs, missing point-slope form or using the wrong point is a common way students lose easy points.

If the graph has a corner, cusp, vertical tangent, or discontinuity, the derivative does not exist there.

5. Multiple Representations of the Derivative

You need to recognize derivatives in four forms.

Graphical

• Positive derivative → function increasing
• Negative derivative → decreasing
• Zero derivative → horizontal tangent

Numerical

Estimate with small difference quotients:
$$\frac{f(3.01)-f(3)}{0.01}$$
Smaller intervals give better approximations.

Analytical

Using the limit definition to produce a formula for $f'(x)$.

Verbal

Interpreting meaning in context:

• “The rate at which water depth is rising at time $t$.”
• “Instantaneous velocity.”

The AP exam loves switching between these without warning. If you truly understand the derivative, the representation doesn’t matter.