Unit 2: Differentiation: Definition and Fundamental Properties

Average rate of change, instantaneous rate of change, and the geometric idea of a derivative

A central theme in calculus is that many real situations involve change, and you often want to describe that change precisely. Before you ever write a derivative symbol, you start with a simpler (and older) idea: average rate of change over an interval.

Average rate of change as a secant slope

If you have a function f and you look at how its output changes between two input values x=a and x=b, the average rate of change (the rate of change over an interval of time/inputs) is

\frac{f(b)-f(a)}{b-a}

This is the usual slope formula from algebra (often stated informally as “y_2-y_1 over x_2-x_1”). Geometrically, it is the slope of the secant line through the two points \bigl(a,f(a)\bigr) and \bigl(b,f(b)\bigr) on the graph.

Why this matters: average rate of change is measurable from data (two measurements give you a change in output over a change in input). But many questions you care about are about what happens “right now” or “at an instant,” not over a whole interval.

Instantaneous rate of change as a tangent slope

The instantaneous rate of change at x=a is the rate at a specific point in time/input. Geometrically, this corresponds to the slope of the tangent line at x=a.

For a straight (linear) line, slope is “rise over run.” For a curved graph, you can’t get a single constant slope that way, so you approximate with a secant line. As the two points on the secant get closer together, the approximation becomes more accurate. In the limit, the secant approaches a tangent line that touches the curve at exactly one point and matches the curve’s direction there.

A typical “moving point” choice is x=a+h, where h is a small change in x. Then the secant slope becomes

\frac{f(a+h)-f(a)}{(a+h)-a}

which simplifies to

\frac{f(a+h)-f(a)}{h}

As h approaches 0, that secant line “rotates” toward the tangent line (when a tangent exists). This limiting secant-slope process is the definition of the derivative.

Difference quotient (the expression you limit)

The expression

\frac{f(a+h)-f(a)}{h}

is called a difference quotient. You should interpret it as “change in f divided by change in x,” with the change in x being h.

Common conceptual pitfall: students sometimes think setting h=0 is the goal. It is not. If you plug in h=0 directly, you get 0/0, which is undefined. The derivative comes from the limit as h\to 0 (as the change in x becomes infinitesimally small), not from substituting h=0.

Worked example: average rate of change

Let f(x)=x^2. The average rate of change from x=2 to x=5 is

\frac{f(5)-f(2)}{5-2}=\frac{25-4}{3}=7

Interpretation: over that interval, f increases by 7 units of output per 1 unit of input, on average.

Worked example: estimating an instantaneous rate from a table

Suppose you have values of f near x=2:

A symmetric estimate for the derivative at x=2 uses points equally spaced around 2, for example 1.99 and 2.01:

f'(2)\approx\frac{f(2.01)-f(1.99)}{2.01-1.99}=\frac{4.0401-3.9601}{0.02}=4

This matches what you would expect if f(x)=x^2, since the true derivative is f'(x)=2x and f'(2)=4.

Exam Focus

• Typical question patterns:
  • “Estimate f'(a) from a table/graph” using nearby values and a difference quotient.
  • “Find the slope of the secant line” between two points and interpret it as an average rate of change.
  • “Explain the meaning of the derivative in context” (units matter).
• Common mistakes:
  • Using a one-sided difference when a symmetric difference is available (often less accurate).
  • Mixing up order in the numerator (doing f(a)-f(a+h) but keeping denominator as h).
  • Forgetting that slope units are “output units per input unit.”

The limit definition of the derivative

The derivative formalizes instantaneous rate of change using limits.

Definition at a point

The derivative of f at x=a is defined by

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

If this limit exists (as a finite real number), then f is differentiable at a.

Why this matters: this definition is the foundation that justifies all derivative rules you learn later. You are expected to understand the derivative as a limit conceptually, and you may be asked to compute a derivative from the definition for simpler functions.

Equivalent limit form

A common equivalent definition uses a second variable approaching the point:

f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}

This is the same idea: the slope of the secant line between x and a approaches the slope of the tangent line.

A practical note: sometimes one form is algebraically cleaner than the other. If you have trouble substituting a+h into a complicated function, the x\to a form may feel simpler.

The derivative as a limit of slopes

It helps to say out loud what the expression means. The numerator f(a+h)-f(a) is the change in output, h is the change in input, the quotient is the average rate of change over a tiny interval, and the limit as h\to 0 attempts to capture the instantaneous rate of change.

If the left-hand and right-hand approaches to 0 give different results (or the value blows up without representing a finite slope), the derivative at that point does not exist.

Worked example: derivative at a point from the definition

Let f(x)=x^2. Find f'(3) using the definition.

Start with

f'(3)=\lim_{h\to 0}\frac{f(3+h)-f(3)}{h}

Compute each piece:

f(3+h)=(3+h)^2=9+6h+h^2

f(3)=9

Substitute:

f'(3)=\lim_{h\to 0}\frac{(9+6h+h^2)-9}{h}=\lim_{h\to 0}\frac{6h+h^2}{h}

Factor out h and cancel (this is the key step that removes the 0/0 form):

f'(3)=\lim_{h\to 0}(6+h)=6

So the slope of the tangent line to y=x^2 at x=3 is 6.

Worked example: using algebra tricks (rationalizing)

Some functions require an algebra move to simplify the difference quotient.

Let f(x)=\sqrt{x}. Find f'(a) from the definition.

f'(a)=\lim_{h\to 0}\frac{\sqrt{a+h}-\sqrt{a}}{h}

This fraction is hard because of the radical. Multiply by the conjugate:

f'(a)=\lim_{h\to 0}\frac{\sqrt{a+h}-\sqrt{a}}{h}\cdot\frac{\sqrt{a+h}+\sqrt{a}}{\sqrt{a+h}+\sqrt{a}}

Then the numerator becomes a difference of squares:

f'(a)=\lim_{h\to 0}\frac{(a+h)-a}{h(\sqrt{a+h}+\sqrt{a})}=\lim_{h\to 0}\frac{h}{h(\sqrt{a+h}+\sqrt{a})}

Cancel h:

f'(a)=\lim_{h\to 0}\frac{1}{\sqrt{a+h}+\sqrt{a}}=\frac{1}{2\sqrt{a}}

This shows the derivative exists for a>0, and it also reveals why \sqrt{x} has a vertical tangent at x=0 (the expression blows up as a\to 0^+).

Exam Focus

• Typical question patterns:
  • “Find f'(a) using the limit definition” for polynomials, roots, or simple rational functions.
  • “Write the limit expression that defines f'(a)” (sometimes no computation required).
  • “Use the derivative definition to show a function is differentiable at a point” (or not).
• Common mistakes:
  • Substituting h=0 before simplifying; you must simplify first, then take the limit.
  • Expanding incorrectly (especially squaring binomials like (a+h)^2 ).
  • Forgetting the conjugate technique when radicals appear.

The derivative as a function and derivative notation

So far, f'(a) is “the derivative at a specific point.” But you can also think of the derivative as a new function that gives a slope for every input where it exists.

From a number to a whole function

If you replace the fixed point a with a variable x in the definition, you get

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

When this limit exists, the output is the slope of the tangent line at that input. So f' is a function whose value at x is the instantaneous rate of change of f at x.

Why this matters: once you have f'(x), you can evaluate it at many points quickly, study where f is increasing or decreasing later in the course, and model rates (like velocity) as functions of time.

Notation: many symbols, same idea (including second derivative notation)

Calculus uses several notations for derivatives. You must be comfortable translating between them.

It’s also common to name second derivatives:

A key conceptual point: \frac{dy}{dx} is not usually treated as an ordinary fraction in Unit 2, but it is extremely useful for expressing “rate of change of output with respect to input.” Later, it behaves fraction-like in a controlled way (especially with the chain rule), but you should not base algebra on that intuition unless you know it is justified.

Graphical meaning of f'

If you look at the graph of f: where f is increasing, tangent slopes are positive, so f'(x)>0; where f is decreasing, tangent slopes are negative, so f'(x)