Chapter 5 · Test 1
Derivatives & Differentials for Economics
Derivative rules, differentials, curvature, and Taylor approximation.
What you will learn
- Differentiate a polynomial
- Calculate marginal revenue
- Compute point elasticity
Worked example
Differentiate a polynomial: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Differentiate a polynomial
Let f(x) = 3x⁴ − 5x² + 2. Find f′(2).
d(cxⁿ)/dx = cnxⁿ⁻¹
In plain English: To differentiate c times x to the nth power, multiply by n and reduce the exponent by one.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- x
- the input variableHow to read: x
- c
- a constant coefficient multiplying a power of xHow to read: c
- n
- the exponent in the power-rule templateHow to read: n
- f(x)
- the original functionHow to read: f of x
- f′(x)
- the Derivative: The local linear rate of change of a function; f′(a) = lim[h→0] (f(a + h) − f(a))/h when this limit exists. of f with respect to xHow to read: f prime of x
Solution
- Differentiate 3x⁴: 12x³.
- Differentiate −5x²: −10x. The constant 2 differentiates to 0.
- Thus f′(x) = 12x³ − 10x.
- Evaluate at x = 2: f′(2) = 12(8) − 10(2) = 96 − 20 = 76.
Visual check
Function and derivative at x = 2
The dashed tangent shows the local slope measured by f′(2).
- f(x)
- Tangent, slope 76
How to read this graph
- Horizontal axis: x
- Vertical axis: f(x)
- Curves and lines: f(x), Tangent, slope 76
- Marked points: Open the exact-coordinate table below to read their values.
What the graph shows: The curve passes through (2, 30). Its tangent there has slope 76.
Read the marked points as data
| Feature | x | f(x) |
|---|---|---|
| (2, 30) | 2 | 30 |
Logic and solving tips
- Apply the power rule term by term, then evaluate the derivative at the requested point.
- Differentiate symbolically before substituting the requested x-value; this prevents mixing the function with its derivative.
Where economists use it
A producer differentiates an estimated cost or production curve to find the extra cost or output created by one more unit of activity.
Key idea: Apply the power rule term by term, then evaluate the derivative at the requested point.
Applied case study
Deriving a marginal pollution-abatement cost curve
Scenario
Environmental economists differentiate total abatement cost to find the cost of removing one additional unit. An EPA procedures manual explicitly defines the marginal cost curve as the first derivative of the total cost curve.
Problem
A teaching facility has C(a) = 100 + 2a² + 0.5a³, where C is thousands of dollars and a is thousands of tons removed. Find marginal abatement cost at a = 4.
MC(a) = C′(a)
In plain English: Marginal cost is the derivative of total cost, so it measures the approximate extra cost of producing one more unit.
Worked solution
- Differentiate the constant: d(100)/da = 0.
- Differentiate the remaining terms: d(2a²)/da = 4a and d(0.5a³)/da = 1.5a².
- Thus MC(a) = 4a + 1.5a².
- Evaluate: MC(4) = 4(4) + 1.5(4²) = 16 + 24 = 40.
Economic interpretation: At 4,000 tons already removed, the modeled marginal cost is 40 thousand dollars per thousand tons, equivalent to about $40 per additional ton.
Source: U.S. EPA: Areawide Assessment Procedures Manual, Volume III The derivative relationship between total and marginal control cost is sourced. The polynomial and coefficients are teaching assumptions, not an EPA estimate.
Reasoning habits that help
- Apply the power rule term by term, then evaluate the derivative at the requested point.
- Marginal revenue is the derivative of total revenue, not simply the market price.
- For a constant-elasticity demand q = Apᵏ, the point elasticity equals the exponent k.
Where economists use these methods
- A producer differentiates an estimated cost or production curve to find the extra cost or output created by one more unit of activity.
- A firm compares the revenue from one additional sale with its marginal cost to decide whether expanding output raises or lowers profit.
- A retailer uses demand elasticity to predict the percentage change in sales after a price change, while a tax analyst uses it to estimate who bears more of a tax.
Chapter vocabulary
Study 41 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Secant line
A line through two distinct points on a function's graph.
Textbook reference: Chapter 5, Section 5.1 Definition of a Tangent Line, printed page 129Average rate of change[f(x_1) - f(x_0)]/(x_1 - x_0)
The change in a function divided by the corresponding change in its input over an interval.
Textbook reference: Chapter 5, Section 5.1 Definition of a Tangent Line, printed page 130Tangent line
The limiting position of secant lines as the second point approaches the point of tangency.
Textbook reference: Chapter 5, Section 5.1 Definition of a Tangent Line, printed pages 128, 132Instantaneous rate of change
The limiting rate at which a function changes at one point, measured by its derivative there.
Textbook reference: Chapter 5, Section 5.1 Definition of a Tangent Line, printed page 128Difference quotient[f(x + h) - f(x)]/h
The secant slope used in the limit definition of a derivative.
Textbook reference: Chapter 5, Section 5.1 Definition of a Tangent Line, printed pages 129, 132Derivativef'(x) = lim_{h→0}[f(x + h) - f(x)]/h
The limit of the difference quotient, giving a function's local rate of change and tangent slope.
How to read: f prime of x
Textbook reference: Chapter 5, Section 5.2 Definition of the Derivative and the Differential, printed page 134First derivative functionf': x ↦ f'(x)
The function assigning to each differentiable input the derivative of the original function at that input.
Textbook reference: Chapter 5, Section 5.2 Definition of the Derivative and the Differential, printed pages 134-135Total differentialdy = f'(x) dx
For a one-variable function, the derivative multiplied by a small input change, giving the linear approximation to the output change.
How to read: d y equals f prime of x times d x
Textbook reference: Chapter 5, Section 5.2 Definition of the Derivative and the Differential, printed pages 136-137Marginal costMC(q) = C'(q)
The rate at which total cost changes as output increases.
Example: It approximates the additional cost of producing one more unit when output is finely divisible.
Textbook reference: Chapter 5, Section 5.2 Definition of the Derivative and the Differential, printed page 138Marginal analysis
Economic analysis that compares the incremental benefits and costs of a small change in an activity.
Textbook reference: Chapter 5, Section 5.2 Definition of the Derivative and the Differential, printed pages 138-139Left-hand derivativef'_-(a)
The limiting difference quotient as the input approaches a point through smaller values.
Textbook reference: Chapter 5, Section 5.3 Conditions for Differentiability, printed page 141Right-hand derivativef'_+(a)
The limiting difference quotient as the input approaches a point through larger values.
Textbook reference: Chapter 5, Section 5.3 Conditions for Differentiability, printed page 141Differentiable function
A function whose derivative exists at every point in the stated domain, with matching one-sided derivatives at interior points.
Textbook reference: Chapter 5, Section 5.3 Conditions for Differentiability, printed pages 141-142Rules of differentiation
Proven formulas that obtain derivatives of common functions and combinations without repeatedly applying the limit definition.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed page 147Power ruled(x^a)/dx = ax^{a-1}
The rule that differentiating x raised to a constant power multiplies by the exponent and lowers the exponent by one.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed pages 147-148Chain ruled f(g(x))/dx = f'(g(x))g'(x)
The rule for differentiating a composite function by multiplying the outer derivative by the inner derivative.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed page 148Product rule(fg)' = f'g + fg'
The rule for differentiating the product of two differentiable functions.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed page 155Quotient rule(f/g)' = (f'g - fg')/g^2
The rule for differentiating a ratio of differentiable functions when the denominator is nonzero.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed page 158Inverse function rule(f^{-1})'(y) = 1/f'(x), y = f(x)
The rule giving an inverse function's derivative as the reciprocal of the original derivative at the corresponding point.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed page 162Marginal revenueMR(q) = R'(q)
The rate at which total revenue changes as quantity sold increases.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed pages 154, 156Marginal product of an inputMP(L) = dq/dL
The rate at which output changes as one input increases, holding the model's other inputs fixed.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed pages 151, 156Average functionA(x) = f(x)/x
A total quantity divided by the number of units over which it is measured.
Example: Average cost is total cost divided by output.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed pages 158-159Arc elasticity
Elasticity measured over a finite movement between two points using percentage changes over that interval.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed pages 167-168Point elasticityε_{y,x} = (dy/dx)(x/y)
The local percentage responsiveness of one variable to another at a specified point.
How to read: the elasticity of y with respect to x
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed page 168Price elasticity of demandε_D = -(dq/dp)(p/q)
The percentage change in quantity demanded divided by the percentage change in own price, holding other demand factors fixed.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed pages 168-169Price elasticity of supplyε_S = (dq/dp)(p/q)
The percentage change in quantity supplied divided by the percentage change in own price, holding other supply factors fixed.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed page 171Constant-elasticity functionq = Ap^{-ε}
A relationship whose point elasticity has the same value throughout its domain.
Textbook reference: Chapter 5, Section 5.4 Rules of Differentiation, printed pages 169-170Higher order derivativef''(x), f^{(n)}(x)
A derivative taken repeatedly, such as the second derivative of a function.
How to read: f double prime of x; the nth derivative of f
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed page 175Second derivativef''(x) = d^2y/dx^2
The derivative of the first derivative, describing how the function's slope changes.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed pages 175-176Convex functionf''(x) ≥ 0
A function that bends no higher than its secant lines; when twice differentiable, a nonnegative second derivative is sufficient and necessary on an interval.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed pages 175, 177Strictly convex function
A convex function that lies strictly below secants between distinct points; a positive second derivative is sufficient.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed page 177Concave functionf''(x) ≤ 0
A function that bends no lower than its secant lines; when twice differentiable, a nonpositive second derivative characterizes it on an interval.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed pages 177-178Strictly concave function
A concave function that lies strictly above secants between distinct points; a negative second derivative is sufficient.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed page 178Point of inflection
A point at which a function changes from convex to concave or from concave to convex.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed pages 178-179Diminishing marginal product
A property in which the marginal product of a variable input falls as more of that input is used, holding other inputs fixed.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed pages 179-180Marginal propensity to consumeMPC = C'(Y)
The increase in consumption associated with a small increase in income.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed page 183Average propensity to consumeAPC = C(Y)/Y
Consumption divided by income at a given income level.
Textbook reference: Chapter 5, Section 5.5 Higher Order Derivatives, printed page 183Taylor series expansion
A polynomial approximation built from a function's value and successive derivatives at a chosen point, plus a remainder.
Textbook reference: Chapter 5, Section 5.6 Taylor Series Formula and the Mean-Value Theorem, printed pages 185-186Remainder term
The part of a Taylor formula that measures the approximation error left after the retained polynomial terms.
Textbook reference: Chapter 5, Section 5.6 Taylor Series Formula and the Mean-Value Theorem, printed pages 185-186Mean-value theoremf'(c) = [f(b) - f(a)]/(b - a)
For a continuous function differentiable inside an interval, some interior point has derivative equal to the interval's average slope.
Textbook reference: Chapter 5, Section 5.6 Taylor Series Formula and the Mean-Value Theorem, printed pages 187-188Linear approximationf(x) ≈ f(x_0) + f'(x_0)(x - x_0)
An estimate near x_0 using the tangent line at x_0.
Textbook reference: Chapter 5, Section 5.6 Taylor Series Formula and the Mean-Value Theorem, printed pages 188-189End-of-chapter problems
Practice all 10 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems