Calculus I · Unit 2A · lesson

The Derivative as a Function

Concept

Learning objectives

Derive a formula for $f'(x)$ from the limit definition and interpret its graph.

The Derivative Function

Explanation

Before the formulas

The symbols in The Derivative as a Function compress three different questions: what the function value is, how two values compare, and what the comparison approaches when the inputs merge. Keep those questions separate. Most confusion at the beginning of differential calculus comes from treating an instantaneous rate as if it were an ordinary quotient over zero distance or zero time. It is not. It is a limit of ordinary quotients over nonzero intervals.

As you work, translate every expression into a sentence. Identify the input, the output, the point of interest, and the units. Then decide whether the problem is asking for a number at one point, a formula for all points, or a line that represents local behavior. This slower reading habit quickly becomes faster than trying to repair symbol errors after several lines of algebra.

Computing $f'(a)$ at one point answers one local question. Repeating the process for every allowable input creates the derivative function $f'$ . Its graph is a compact record of all the slopes of $f$ : height on the $f'$ -graph corresponds to slope on the $f$ -graph.

This translation is one of the most important habits in calculus. When $f'$ is positive, $f$ rises; when $f'=0$ , $f$ has a horizontal tangent; and when $|f'|$ is large, $f$ changes rapidly. Later units will extract an astonishing amount of global information from this local slope function.

A derivative at one point is a number. If we repeat the calculation for every input where the derivative exists, the resulting values form a new function.

Definition

Derivative function

The derivative function of $f$ is

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

wherever this limit exists.

Guided walkthrough

Build the derivative of $x^2$

Find a formula for the derivative of $f(x)=x^2$ .

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Concept

Heights on $f'$ are slopes on $f$

If $f'(2)=4$ , then the point $(2,4)$ lies on the graph of $f'$ , and the tangent to $f$ at input $2$ has slope $4$ . A derivative graph is a slope record of the original graph.

**Visual study**
For $f(x)=x^2$ , the derivative graph $f'(x)=2x$ records the slope of the parabola.
For f(x)=x², the derivative graph f'(x)=2x records the slope of the parabola.

Exercise

Use the limit definition to find the derivative function of $f(x)=3x+5$ .

Exercise

Use the limit definition to find the derivative function of $f(x)=x^2-6x$ .

Exercise

If a graph of $f$ is increasing and becoming steeper, what signs should $f'$ and its trend have?

Exercise

Sketch a possible graph of $f'$ when $f$ is a line with slope $-3$ .

Application

Fuel use and speed sensitivity

Suppose a vehicle's fuel use over a fixed route is modeled by

F(v)=4+\frac{(v-55)^2}{900}

gallons when the average speed is $v$ miles per hour. Then

F'(v)=\frac{v-55}{450}.

At $v=70$ , $F'(70)=1/30$ gallon per mph. Near $70$ mph, increasing speed by one mph raises predicted fuel use by about $0.033$ gallon. At $v=55$ , the derivative is zero, indicating the locally most fuel-efficient speed in this model.

Optional advanced note

Advanced note: derivative functions can be surprisingly irregular

A derivative need not be continuous. For example, a carefully chosen function can be differentiable at every point while its derivative oscillates wildly near one point. Yet derivatives are not completely arbitrary: they satisfy the intermediate value property known as Darboux's Theorem. A derivative cannot jump directly from one value to another without taking every value in between, even when the derivative itself is discontinuous.

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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