DPST1013 Chap.7 Differentiation and the Mean Value Theorem
Differentiation and the Mean Value Theorem
The derivative is developed from a limit of secant slopes, then calculated efficiently through product, quotient and chain rules. Implicit differentiation and related rates keep dependencies visible. The Mean Value Theorem connects average and instantaneous change, while L'Hopital addresses specific indeterminate limit forms.
What this chapter covers
- 01
Derivative from first principles
- 02
Product, quotient and chain rules
- 03
Implicit differentiation
- 04
Rates and higher derivatives
- 05
Mean Value Theorem
- 06
L'Hopital and optimisation
Worked example · free
Find the Mean Value Theorem point
- 1As a polynomial, the function has no breaks across the interval and has a derivative at every interior point.
- 1The endpoint secant slope is zero.
- 1Solve f prime(c) = 3c squared - 3 = 0 to get c = plus or minus 1.
- 1Only c = 1 lies in the open interval (-1, 2).
Key terms
- Derivative
- The limit of secant slopes, interpreted as instantaneous rate or tangent slope.
- Chain rule
- A rule differentiating a composition by multiplying the outer derivative by the inner derivative.
- Critical number
- A domain point where the derivative is zero or does not exist.
- Mean Value Theorem
- A theorem matching an interior tangent slope to the endpoint average slope under continuity and differentiability conditions.
Differentiation and the Mean Value Theorem FAQ
Why must I state the Mean Value Theorem conditions?
The conclusion is guaranteed only when the function has no discontinuity between the included endpoints and has a derivative at every interior point. Solving the derivative equation without those facts does not justify the existence claim.
When can I use L'Hopital's rule?
First substitution must produce a suitable zero-over-zero or infinity-over-infinity form, with the required differentiability conditions nearby. Differentiate numerator and denominator separately, then reassess the resulting limit rather than assuming one application finishes it.
How do I avoid missing a chain-rule factor?
Name the inner expression temporarily, differentiate the outer function while leaving the inner expression in place, then multiply by the inner derivative. A quick numerical slope or symbolic check can expose a missing constant factor.
How is differentiability related to continuity?
Differentiability at a point implies continuity there, because a finite derivative controls the local change in the function. The converse is false: an absolute-value corner is continuous but has unequal one-sided slopes. When a derivative is requested at a join, first check continuity and then compare the left and right derivative limits. A continuous graph alone does not guarantee a tangent slope.
What is the safest order for implicit differentiation?
Differentiate both sides term by term with respect to the independent variable, attaching a derivative factor whenever a term depends on y. Collect every term containing the requested derivative on one side, factor it, and divide only at the end. Substitute a point after deriving the general expression unless early substitution is explicitly justified, because substituting too soon can erase necessary variable dependence.
How do I identify all candidates in an optimisation problem?
Write the objective function on its feasible domain, then find interior points where its derivative is zero or undefined. Include relevant endpoints and any boundary created by the constraints. Evaluate or compare the objective at every candidate and interpret the winning value in context. A derivative equation locates candidates; it does not by itself prove an absolute maximum or minimum.
Why do related-rates calculations need units and signs?
Each derivative describes change with respect to time, so its units distinguish lengths per second from areas or volumes per second. The sign records whether a quantity is increasing or decreasing. Differentiate the relationship before inserting the instant's numerical values, then solve for the requested rate. A negative result may be physically correct, but the concluding sentence should explain its direction.
What information does the second derivative provide?
Its sign describes concavity where the function is twice differentiable: positive bends upward and negative bends downward. A sign change can confirm an inflection point, but merely solving the second derivative equal to zero produces candidates only. The second-derivative test may also classify a stationary point when its value there is non-zero. Combine this evidence with the first derivative and domain information.
Exam move
Derive one simple derivative from first principles so the rules retain meaning, then drill product, quotient and chain structures separately. For implicit work, attach dy/dx to every differentiated y-term and collect those terms before substitution. For theorem questions, state hypotheses and the eligible interval. For optimisation, include undefined derivative points and boundaries.
Check related rates with units and verify signs against the physical direction described. Classify an expression's outermost operation before selecting a rule, and mark inner functions with brackets. After differentiating, compare the result with the original function's scale, domain and likely sign. Build a theorem checklist that separates hypotheses, existence conclusions and equations used to locate points.
For application problems, define variables and units before forming a relationship, then translate the final derivative back into a sentence. Redo mixed questions without a formula list so recognition, not just execution, becomes automatic.
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