Differentiation: Definition and Fundamental Properties

Average vs. Instantaneous Rate of Change (and the Tangent Line Problem)

What “rate of change” means in calculus

A rate of change describes how one quantity changes as another quantity changes. If you have a function f, you can think of f(x) as an output that depends on an input x. The rate of change tells you how sensitive the output is to small changes in the input.

In Algebra, you may have seen slope as “rise over run.” In calculus, slope becomes more powerful because we can talk about the slope at a single point on a curve, not just between two points on a line.

There are two closely related ideas:

  • Average rate of change: change over an interval.
  • Instantaneous rate of change: change at an instant (at a point).

The derivative will formalize instantaneous rate of change.

Average rate of change and secant lines

The average rate of change of f from x=a to x=b is

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

A common “algebra slope” way to say the same thing is

\frac{y_2-y_1}{x_2-x_1}

Geometrically, this average rate of change is the slope of the secant line through the points \bigl(a,f(a)\bigr) and \bigl(b,f(b)\bigr).

Why it matters: average rate of change is how you estimate speed from “distance traveled over time,” and it is the stepping stone to instantaneous rate of change because you can “zoom in” by shrinking the interval.

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From secant slope to tangent slope (the key limiting idea)

Suppose you want the slope of the curve at x=a. You cannot compute slope “at a point” using just one point; you need two points. The calculus idea is to use a second point very close to a.

Let the second point be at a+h, where h is a small number (positive or negative). The slope of the secant line through \bigl(a,f(a)\bigr) and \bigl(a+h,f(a+h)\bigr) is

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

As h approaches 0, the second point approaches the first point. If the slopes of these secant lines approach a single value, that limiting value is the slope of the **tangent line** at x=a. The closer the points are, the more accurate the secant-line approximation becomes.

This is the heart of differentiation: instantaneous rate of change is a limit of average rates of change.

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Worked example: average rate of change

Let f(x)=x^2. Find the average rate of change from x=1 to x=3.

Compute outputs: f(3)=9 and f(1)=1. Then

\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4

Interpretation: over the interval from input 1 to input 3, the output increases on average by 4 units per 1 unit of input.

Worked example: a secant slope expression that leads to a tangent slope

For f(x)=x^2, write the secant slope from x=a to x=a+h.

Start with

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

Compute

f(a+h)=(a+h)^2=a^2+2ah+h^2

So

\frac{f(a+h)-f(a)}{h}=\frac{(a^2+2ah+h^2)-a^2}{h}=\frac{2ah+h^2}{h}=2a+h

You can already see the limit as h\to 0 will be 2a, which becomes the derivative at a.

Exam Focus
  • Typical question patterns:
    • Compute an average rate of change from a table, graph, or formula.
    • Find the slope of a secant line and then use a limit idea to describe a tangent slope.
    • Interpret a rate of change in context (include units).
  • Common mistakes:
    • Swapping a and b inconsistently (this flips the sign). Keep numerator and denominator in matching order.
    • Using b-a in the denominator but f(a)-f(b) in the numerator.
    • Treating “average rate of change” as “instantaneous” without a limiting process.

The Derivative Defined as a Limit

What the derivative is

The derivative of a function at a point measures the function’s instantaneous rate of change at that point. Geometrically, it is the slope of the tangent line to the graph at that point (when the tangent line exists in the usual sense).

The derivative is defined using a limit because you approximate the tangent slope by secant slopes and then let the interval shrink to zero.

The (difference quotient) definition at a point

The derivative of f at x=a is

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

Key pieces to understand: h represents a small change in the input; f(a+h)-f(a) is the resulting change in output; dividing by h gives an average rate of change over the small interval; and the limit as h\to 0 captures the instantaneous rate of change.

A common equivalent form uses a second input value x approaching a:

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

Both definitions describe the same idea; which one you use often depends on convenience.

When the derivative exists

The derivative at a exists if the limit exists and is finite.

Graph connections you should recognize:

  • At a smooth point, slopes of secant lines from both sides approach the same number.
  • At a corner or cusp, the left-hand and right-hand slopes do not match (or blow up differently).
  • At a vertical tangent, the slope tends to infinity (the derivative is not a finite real number).
  • At a discontinuity, the derivative cannot exist.

Worked example: derivative from the limit definition

Find f'(a) for f(x)=x^2 using the definition.

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

Since

(a+h)^2=a^2+2ah+h^2

we get

\frac{(a+h)^2-a^2}{h}=\frac{a^2+2ah+h^2-a^2}{h}=\frac{2ah+h^2}{h}=2a+h

So

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

and the derivative function is f'(x)=2x.

Worked example: a function with a non-smooth point

Consider f(x)=|x| at x=0.

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

If h>0, then \frac{|h|}{h}=1. If h