ECON1202 Chap.10 Differentiation and Elasticity
Differentiation and Elasticity
Differentiation measures local change, but the correct rule depends on expression structure. Constants and powers use basic rules. Products require the sum of two contributions because both factors change. Quotients use a signed numerator divided by the squared denominator, and the order of its terms must be preserved. Compositions use the chain rule by multiplying the outer derivative by the derivative of the inner function.
Exponential and logarithmic functions retain recognisable forms, with the inner derivative included when their arguments are composite. Simplifying before differentiating can reduce work, provided domain restrictions are preserved and invalid cancellation across addition is avoided. Point elasticity scales a demand derivative by the ratio of price to quantity, producing a unit-free proportional response.
Its sign should match the demand slope, and its interpretation is local to the selected point. A complete solution names the function structure, shows the general derivative before substitution, retains any original domain restriction and interprets the units, sign and magnitude of the result.
What this chapter covers
- 01
Constant and power rules
- 02
Exponential derivatives
- 03
Logarithmic derivatives
- 04
Product rule
- 05
Quotient rule
- 06
Chain rule
- 07
Derivative units
- 08
Point elasticity
Worked example · free
Differentiate a composite exponential
- +1Identify the outer exponential and inner function g(x)=x^2-3x.
- +1Differentiate the inner function: g'(x)=2x-3.
- +1Apply the chain rule: f'(x)=(2x-3)e^(x^2-3x).
- +1At x=2, f'(2)=e^(-2), approximately 0.1353.
Key terms
- Derivative
- The local rate of change of an output with respect to an input.
- Product rule
- Derivative rule that adds the change contribution from each factor of a product.
- Quotient rule
- Derivative rule for a ratio that preserves numerator and denominator order.
- Chain rule
- Rule that multiplies derivatives along a sequence of nested dependencies.
- Marginal change
- The derivative interpreted as a small-output response per input unit.
- Point elasticity
- A unit-free local proportional response formed by scaling a derivative.
Differentiation and Elasticity FAQ
How do I choose the right derivative rule?
Inspect how the expression is assembled after valid simplification. A product of changing factors needs the product rule, a ratio needs the quotient rule, and an outer function applied to an inner function needs the chain rule. Several rules may be nested.
Why does the quotient-rule order matter?
The numerator is f prime times g minus g prime times f. Reversing those terms changes the sign. Label the numerator and denominator functions first, retain parentheses, and keep the original restriction that the denominator cannot be zero.
What makes elasticity different from a derivative?
A derivative has units of output per input unit. Point elasticity multiplies by input divided by output, removing units and describing a local percentage response. Its sign and magnitude should be interpreted at the selected point.
How can a derivative be checked?
Use algebraic structure, units and a small numerical secant. A product derivative should contain contributions from both factors, a chain derivative should include the inner derivative, and the secant slope near a point should be close to the calculated derivative.
What should a structured differentiation solution make explicit?
Read the expression tree before applying a rule. Label the outer and inner functions in a composition, both moving factors in a product, or the numerator and denominator in a quotient. Write the rule symbolically, differentiate each named part and substitute it back before simplifying. Preserve the original domain when cancelling factors, especially around denominators and roots.
For a stationary point, solve the derivative equation and check that every candidate lies in the domain. For point elasticity, evaluate the derivative and function at the same point, scale by input divided by output and state the sign convention. Interpret sign as direction and absolute magnitude as responsiveness.
Verify the algebra with a nearby secant slope, a known special case or substitution into the unsimplified derivative structure. A polished final line names the marginal or elasticity quantity and its evaluation point rather than reporting an unlabelled number. When a result is simplified, keep one unsimplified line so a missing chain factor or reversed quotient numerator remains easy to audit.
Exam move
Create a rule-selection routine: simplify safely, circle products and quotients, underline inner functions, then differentiate from the outside structure inward. Practise product and quotient rules with functions that could tempt an incomplete one-factor derivative. For chains, write the inner derivative on a separate line and verify it appears as a factor in the final result.
Preserve domain restrictions when simplification cancels a factor. Evaluate only after deriving the general formula so the method remains visible. For point elasticity, calculate quantity and the demand derivative separately before substituting into price divided by quantity times the derivative. Check sign against the slope and describe the result as a local proportional response.
Use a small symmetric numerical slope around selected points to catch missing terms without replacing the symbolic work.
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