Applications of Differentiation — MathsTricks & Calculus

§ 01 What You Will Learn From first derivative to real-world problem solving

Differentiation gives you a tool. Applications of differentiation teach you how to wield it. By the end of this section you will be able to locate and classify critical points, sketch any differentiable function from scratch, solve constrained optimisation problems, relate rates that change together, and apply the most important theorems in calculus.

Key Results at a Glance

f′(c) = 0 or undefined Critical point condition

f′′(c) > 0 → local min Second derivative test

f′′(c) = 0 → inflection? Concavity change

f′(c) = (f(b)−f(a))/(b−a) Mean Value Theorem

L(x) = f(a) + f′(a)(x−a) Linear approximation

xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) Newton's Method

Where Applications of Differentiation Sits

This section bridges pure differentiation and integration. You need a solid command of derivatives before starting, and the skills you develop here (especially curve analysis) are essential preparation for the Fundamental Theorem of Calculus.

Limits

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Differentiation Rules

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Applications of Differentiation

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Integration

§ 02 All Topics Choose where to start — each page is self-contained

Core Topic · Start Here

Maxima & Minima

The centrepiece of differential calculus applications. Learn to find and classify critical points using the first and second derivative tests, distinguish local from global extrema, and apply the Extreme Value Theorem on closed intervals. Indispensable for everything that follows.

Critical Points First Derivative Test Second Derivative Test EVT Global vs Local

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Core Topic

Curve Sketching

Synthesise everything about a function — domain, intercepts, asymptotes, increasing/decreasing intervals, concavity, and inflection points — into an accurate hand-drawn sketch. The seven-step method makes any curve tractable.

Increasing / Decreasing Concavity Inflection Points Asymptotes

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Applied Topic

Optimisation

Translate word problems into calculus. Maximise revenue, minimise cost, find the largest box from a sheet of cardboard. A systematic five-step process — define, constrain, differentiate, test, conclude — handles virtually every exam problem.

Constrained Optimisation Word Problems Closed Interval Method

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Applied Topic

Implicit Differentiation Chain Rule Rates of Change

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Theoretical Foundation

Mean Value Theorem

One of the most important theorems in calculus: if f is continuous on [a, b] and differentiable on (a, b), then at some interior point the instantaneous rate of change equals the average rate of change over the interval. Includes Rolle's Theorem as a special case.

Rolle's Theorem MVT Existence Theorems

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Core Topic

Linear Approximation & Differentials

Near any point where a function is differentiable, it is well-approximated by its tangent line. Learn to use this linearisation for estimation, error analysis, and as the conceptual foundation of Newton's Method and numerical calculus.

Tangent Line Approx Differentials Error Estimation

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Numerical Method

Newton's Method

An iterative root-finding algorithm: starting from an initial guess x₀, repeatedly apply xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) to converge to a root of f. Converges quadratically when it converges — but beware of failure cases.

Root Finding Iteration Numerical Calculus

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Limits & Differentiation

L'Hôpital's Rule

When a limit produces an indeterminate form (0/0 or ∞/∞), differentiate numerator and denominator separately and retry. Covers all seven indeterminate forms with careful algebraic manipulation.

0/0 Forms ∞/∞ Forms 0·∞, ∞−∞, 0⁰

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§ 03 Topic Overview Difficulty level, prerequisites, and what each topic unlocks

Topic	Level	Key Tool	Unlocks
Maxima & Minima	Core	First & Second Derivative Tests	Optimisation, Curve Sketching
Curve Sketching	Core	f′ and f″ sign analysis	Visualising any function
Optimisation	Applied	Critical points + constraints	Real-world problem solving
Related Rates	Applied	Implicit differentiation w.r.t. t	Physics & engineering problems
Mean Value Theorem	Core	Existence theorem	FTC proof, L'Hôpital
Linear Approximation	Core	Tangent line L(x)	Newton's Method, error analysis
Newton's Method	Advanced	Iterative refinement	Numerical computing
L'Hôpital's Rule	Advanced	Differentiating limits	All indeterminate forms
Extreme Value Theorem	Core	Continuity guarantee	Rigorous optimisation proofs

§ 04 Suggested Study Path For first-year university calculus or A-Level Further Maths

Applications of differentiation is best studied in two passes. The first pass covers the three core topics that every calculus student must master: maxima & minima, curve sketching, and basic optimisation. These topics are tightly linked and reinforce each other, understanding critical points makes curve sketching mechanical, and curve sketching makes optimisation intuitive.

The second pass adds theoretical depth and applied breadth: the Mean Value Theorem underpins the rigour of everything else; related rates develops your ability to translate geometry into calculus on the fly; linear approximation connects differentiation to numerical methods; and Newton's Method shows that the tangent line is not just a sketch tool but a computational engine.

A Note on Order Maxima & Minima is the entry point — begin there regardless of anything else. Related Rates and Optimisation can be studied in either order after you are confident with critical points. The MVT and Linear Approximation should come after you have seen several worked examples of curve sketching. Newton's Method comes last and requires Linear Approximation as direct preparation.

If You Are Short on Time

For a single-semester exam revision: prioritise Maxima & Minima, Optimisation, and Curve Sketching, these three topics account for the majority of marks on standard university calculus exams. Related Rates and the MVT are the next tier. Newton's Method, L'Hôpital's Rule, and the EVT are important but often examined less frequently.

§ 05 The Big Theorems Every result in this section flows from one of these four

Applications of differentiation is organised around four cornerstone theorems. Understanding these at a conceptual level (not just mechanically) transforms your ability to solve problems you have never seen before.

Four Cornerstone Theorems

Extreme Value Theorem A continuous function on [a,b] attains its max and min. Guarantees the existence of extrema before we look for them.

Fermat's Theorem If f has a local extremum at c and f is differentiable there, then f′(c) = 0. This is why we set f′ = 0 to find candidates.

Rolle's Theorem If f(a) = f(b), there exists c ∈ (a,b) with f′(c) = 0. The special case that motivates the MVT.

Mean Value Theorem There is always a point where the tangent slope equals the secant slope. Consequence: if f′ > 0 everywhere, f is increasing.

These four theorems form a logical chain: the EVT guarantees extrema exist, Fermat's Theorem says where to look for them (at critical points), Rolle's Theorem is the bridge to the MVT, and the MVT is used to prove the relationship between the sign of f′ and monotonicity: which drives the entire first derivative test.

§ 06 Related Sections

Applications of differentiation sits at the heart of calculus. These sections are natural companions: