DPST1013 Chap.7 Differentiation and the Mean Value Theorem
Differentiation and the Mean Value Theorem
Differentiation is more than a collection of algebraic rules. Its definition connects a limiting difference quotient to a local rate of change; the rules then make that quantity practical to calculate. Reading the outer structure of an expression tells you whether a product, quotient or chain rule is needed. Implicit differentiation extends the method to curves described by an equation instead of a single explicit formula.
The mean value theorem links local rates to a change across an interval. Its hypotheses explain why a derivative bound can control an output difference and why a positive derivative implies an increasing function on an interval. These conclusions become useful in approximation, root uniqueness and the interpretation of a curve.
A theorem name alone is not a substitute for checking continuity, differentiability and the interval. Applications also require precision about what is being sought. A stationary point is an input where the derivative is zero; a local extremum concerns nearby values; an absolute extremum compares the whole allowed domain. On a closed interval, endpoints belong in the candidate comparison.
A second derivative can classify some stationary points, but an inconclusive result calls for further analysis. The example below uses a polynomial to show why a complete answer may need several locations, even when only one minimum value and one maximum value are requested.
What this chapter covers
- 01
Difference quotients and the distinction between continuity and differentiability
- 02
Product, quotient and chain rules, with implicit and higher derivatives
- 03
Mean value hypotheses, derivative bounds and monotonicity conclusions
- 04
Stationary points, nonsmooth candidates, endpoints and absolute extrema
- 05
L’Hôpital’s rule as a conditional limit method and tangent approximation
An absolute extremum can occur in more than one place
- 1The polynomial is continuous on the closed interval [0,4], so absolute extrema are attained. It is differentiable throughout, so the candidates are the interval endpoints and interior stationary points.
- 2Calculate g′(x)=3x²−12x+9=3(x−1)(x−3). Its zeros are x=1 and x=3, both inside the interval. Add x=0 and x=4, giving the complete candidate set {0,1,3,4}.
- 2Evaluate each candidate using the original polynomial: g(0)=0; g(1)=1−6+9=4; g(3)=27−54+27=0; g(4)=64−96+36=4. The repeated values are meaningful and must be retained.
- 2The absolute minimum is 0, attained at x=0 and x=3. The absolute maximum is 4, attained at x=1 and x=4. The derivative is positive before 1, negative between 1 and 3, and positive after 3, which independently agrees with the candidate comparison.
Key terms
- Difference quotient
- A difference quotient divides a function’s output change by the corresponding nonzero input change. Its limit, when it exists as that change tends to zero, defines the derivative. The denominator is nonzero during the quotient calculation even though it approaches zero in the limiting process.
- Chain rule
- The chain rule differentiates a composition by multiplying the derivative of the outer function, evaluated at the inner function, by the derivative of the inner function. The inner factor records how quickly the intermediate input changes and cannot be dropped merely because the outer derivative is familiar.
- Implicit differentiation
- Implicit differentiation finds a derivative from a relation connecting dependent and independent variables. Treat the dependent variable as a local function, so its derivative appears through chain and product rules. After collecting derivative terms, division requires the coefficient being divided by to be nonzero at the point.
- Stationary point
- A stationary point is a point in the domain where the derivative is zero. That condition identifies a candidate for some extrema but does not classify it. A horizontal tangent can occur without a local maximum or minimum, so nearby behaviour or additional derivative information is needed.
- Absolute maximum
- An absolute maximum is an attained function value at least as large as every other value on the specified domain. There may be several inputs attaining the same maximum. The domain qualification matters: changing the interval can change both the largest value and its location.
- Concavity
- Concavity describes how tangent slopes change as the input increases. Where a second derivative exists and is positive, the first derivative increases and the curve is concave up; a negative second derivative gives concavity down. An inflection involves a change of concavity rather than simply a zero second derivative.
- Linear approximation
- A linear approximation uses the tangent expression near a chosen input. It matches the function value and first derivative at that input, giving a local estimate at nearby values. It is generally approximate away from the base point, and its reliability depends on the behaviour of the function nearby.
Differentiation and the Mean Value Theorem FAQ
Can a continuous function fail to have a derivative?
Yes. Continuity requires nearby values to approach the actual value, while differentiability imposes a compatible limiting slope. At a corner, the left and right slopes may disagree even though both pieces meet. For a piecewise join, check the matching values first and then compare the one-sided derivatives; a continuous join alone is insufficient.
Why are endpoints checked in absolute-extremum problems?
An endpoint belongs to a closed interval and can have a value larger or smaller than every interior value. The usual zero-derivative condition for a smooth interior extremum does not require the endpoint slope to vanish. Build the candidate list from endpoints, interior stationary points and any relevant points where differentiability fails, then compare actual function values.
Does a zero second derivative show an inflection?
A zero second derivative is a candidate signal, not a complete classification. To identify an inflection, establish a change of concavity at the appropriate point of the curve. If the second derivative test at a stationary point gives zero, inspect signs of the first derivative or compare nearby values rather than assigning a maximum or minimum automatically.
How does the mean value theorem justify increasing behaviour?
For any two ordered inputs in an interval where the hypotheses hold, the theorem expresses their average rate of change as a derivative at an interior point. If all such derivative values are positive, the output difference has the same positive direction as the input difference. A gap in the domain prevents applying that closed-interval argument across the gap.
When should I avoid L’Hôpital’s rule?
Avoid applying it before identifying an appropriate indeterminate quotient and checking its hypotheses. A quotient with a finite nonzero denominator often allows direct substitution. A product or power may need a justified transformation first. Differentiating numerator and denominator for this limit method is also distinct from using the quotient rule to differentiate the whole function.
Exam move
Practise naming the outer operation before writing a derivative. After a product or chain calculation, expand a simple example independently to check the result. For implicit curves, verify that the requested point lies on the curve before substituting it into the derivative equation; a tangent at an unrelated point is meaningless. Alternate routine differentiation with interpretation.
Use a derivative sign chart to explain increasing and decreasing intervals, then compare an extremum classification with actual candidate values. When applying a theorem, include its interval and hypotheses in the written solution. Finish by matching the output to the question: a derivative formula, a tangent line, a set of extremal locations and a limiting value are different mathematical objects.
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