Calculus I · 2A · lesson
The Derivative as a Function
Learn the derivative as a function with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Derivative meaning and foundationsWhat this section is building
Learn the derivative as a function with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
A derivative exists when shrinking two-point slopes settle to one finite local slope.
Choose whether the task asks for a value at one point, a full derivative function, or an estimate from data.
Confusing the graph's height with its slope or assuming continuity automatically gives differentiability.
Learning objectives
Derive a formula for from the limit definition and interpret its graph.
The Derivative Function
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 at one point answers one local question. Repeating the process for every allowable input creates the derivative function . Its graph is a compact record of all the slopes of : height on the -graph corresponds to slope on the -graph.
This translation is one of the most important habits in calculus. When is positive, rises; when , has a horizontal tangent; and when is large, 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.
Derivative function
The derivative function of is
wherever this limit exists.
Build the derivative of
Find a formula for the derivative of .
Worked solution
Write a real attempt before opening the supplied answer.
Heights on are slopes on
If , then the point lies on the graph of , and the tangent to at input has slope . A derivative graph is a slope record of the original graph.
Read this graph as text
original function and derivative graph. For (f(x)=x 2 ), the derivative graph (f'(x)=2x ) records the slope of the parabola. Teach the derivative function as a record of tangent slopes: slope on the original graph becomes height on the derivative graph.
The visual uses labeled positions, solid and dashed line styles, and written descriptions so original function and derivative graph does not depend on color.
Why it matters: Teach the derivative function as a record of tangent slopes: slope on the original graph becomes height on the derivative graph.
For , the derivative graph records the slope of the parabola.
For f(x)=x², the derivative graph f'(x)=2x records the slope of the parabola.
Use the limit definition to find the derivative function of .
Use the limit definition to find the derivative function of .
If a graph of is increasing and becoming steeper, what signs should and its trend have?
Sketch a possible graph of when is a line with slope .
Fuel use and speed sensitivity
Suppose a vehicle's fuel use over a fixed route is modeled by
gallons when the average speed is miles per hour. Then
At , gallon per mph. Near mph, increasing speed by one mph raises predicted fuel use by about gallon. At , the derivative is zero, indicating the locally most fuel-efficient speed in this model.
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.
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