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Calculus I · Unit 2A · lesson

The Derivative at a Point

Concept

Learning objectives

Use the limit definition to calculate f′(a)f'(a); interpret the result as a rate and a tangent slope.

The Formal Definition at a Point

Explanation

Before the formulas

The main idea behind The Derivative at a Point is local change. A graph may be complicated over a large interval and still behave in a simple, nearly linear way near one input. The derivative records that local direction. It does not describe the total amount of the function, and it does not automatically describe what happens far from the point.

Use three representations whenever possible: a numerical rate from nearby values, a slope on a graph, and a symbolic limit or derivative. When all three tell the same story, the calculation is much easier to trust. When they disagree, the disagreement usually exposes a dropped sign, a misread unit, or a function that is not differentiable at the point.

The derivative exists when the graph near a point is captured, to first order, by one line. The right panel magnifies the marked neighborhood from the left panel.
Read this graph as text

A curve becomes line-like under magnification. The derivative exists when the graph near a point is captured, to first order, by one line. The right panel magnifies the marked neighborhood from the left panel. The left panel shows a visibly curved parabola and its tangent line at x=2 . The red rectangle marks the small region enlarged on the right. In that magnified region, the curve and tangent are almost indistinguishable. This is the practical meaning of differentiability: the line captures the first-order behavior near the point even though it does not match the curve globally.

The visual uses labeled positions, solid and dashed line styles, and written descriptions so a curve becomes line-like under magnification does not depend on color.

Why it matters: This visual should carry more conceptual weight than a standard tangent-line picture. Students often interpret a tangent as a line that "touches once" or as an arbitrary line placed against a curve. The paired views instead show that a derivative is a local approximation statement. The tangent is privileged because its error becomes small relative to the horizontal displacement.

Visual study

The derivative exists when the graph near a point is captured, to first order, by one line. The right panel magnifies the marked neighborhood from the left panel.

At a single point, the graph does not provide two distinct points from which to calculate a slope. The derivative solves that problem by using nearby secant slopes and asking whether they settle toward one number. The resulting value belongs to one input, so f′(a)f'(a) is a number, not yet a new formula.

This lesson is worth doing slowly. Once the difference quotient has been simplified correctly, the limit is often easy. Most errors happen earlier: parentheses are dropped, a common factor is not exposed, or f(a+h)f(a+h) is evaluated as though the +h+h were decorative.

Definition

Derivative at a point

Let ff be defined near x=ax=a. The derivative of ff at aa is

f′(a)=lim⁡h→0f(a+h)−f(a)h,\boxed{f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}},

provided the limit exists as a finite real number.

The derivative has three equivalent interpretations:

• the instantaneous rate of change of ff with respect to xx at aa; • the limiting value of average rates on intervals shrinking toward aa; • the slope of the tangent line to y=f(x)y=f(x) at (a,f(a))(a,f(a)).

Guided walkthrough

Derivative of a quadratic at one point

Use the limit definition to find the derivative of

f(x)=x2−4x+1f(x)=x^2-4x+1

at x=3x=3.

Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Worked example

A reciprocal function from first principles

Find f′(2)f'(2) for f(x)=1/xf(x)=1/x.

Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Common mistake

The symbol hh is not the derivative. It is the change in the input. The derivative is the limit of the entire quotient as h→0h\to0. Cancelling a factor of hh is permitted only after algebra reveals that factor and only because the quotient is studied for h≠0h\ne0.

Exercise

Use the definition to find f′(1)f'(1) for f(x)=x2+2xf(x)=x^2+2x.

Exercise

Use the definition to find g′(−1)g'(-1) for g(x)=3x2−xg(x)=3x^2-x.

Exercise

Use the definition to find p′(4)p'(4) for p(x)=xp(x)=\sqrt{x}. Rationalize the numerator.

Exercise

Explain in one sentence what C′(100)=4.25C'(100)=4.25 means if C(q)C(q) is production cost in dollars for qq units.

Optional advanced note

Analysis preview: differentiability is an error statement

In a later real-analysis course, the derivative is often expressed by

f(a+h)=f(a)+f′(a)h+r(h),r(h)h⟶0.f(a+h)=f(a)+f'(a)h+r(h), \qquad \frac{r(h)}{h}\longrightarrow0.

The remainder r(h)r(h) may not be zero, but it becomes negligible compared with hh. This formulation says more than "the tangent slope exists": it says the tangent line captures the entire first-order behavior of the function. The derivative is therefore the coefficient of the best local linear model.

After the explanation

Use the section idea

Reading lens

Watch a secant slope stabilize into a tangent slope and then generalize from one point to a derivative function.

Mental model

A derivative exists when shrinking two-point slopes settle to one finite local slope.

Decision

Choose whether the task asks for a value at one point, a full derivative function, or an estimate from data.

Common trap

Confusing the graph's height with its slope or assuming continuity automatically gives differentiability.

Check yourself

Can you move among a limit definition, tangent slope, graph estimate, and units without changing the meaning?

Source & rights

Original instruction with traceable references.

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