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QCE · QCE Units 3 & 4

Mathematical Methods

Functions, calculus and statistics — full Paper 1 (tech-free) + Paper 2 (tech-active) practice External Assessments with worked solutions.

20full-length model exams with mark-by-mark answer guides
20detailed note sets — ~120 pages across every topic
64exam-style practice questions with worked solutions
60flashcards for every key term & formula
Allofficial past papers

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Sample revision note

Exponential and Logarithmic Derivatives

1. Foundations: Index and Logarithmic Laws as Differentiation Tools

Before differentiating exponential and logarithmic functions, it is essential to be fluent in manipulating expressions using index and logarithmic laws. These laws are not merely algebraic tools — they actively simplify functions into forms that are far easier to differentiate, and they allow us to verify or rearrange results once derivatives have been found.

Key index laws used in this topic:

  • Product rule (indices): am · an = am+n
  • Quotient rule (indices): am / an = am−n
  • Power law: (am)n = amn
  • Negative index: a−n = 1/an

Key logarithm laws (for base e, i.e. natural logarithm ln):

  • Log of a product: ln(ab) = ln a + ln b
  • Log of a quotient: ln(a/b) = ln a − ln b
  • Log of a power: ln(an) = n ln a
  • Inverse relationship: eln x = x and ln(ex) = x for x > 0

These laws become especially powerful when applied before differentiating. For example, ln(x3) simplifies to 3 ln x, making the derivative immediately 3/x rather than requiring the chain rule on a composite expression. Similarly, expressions like e3x + 1 · ex can be combined into e4x + 1 before differentiating.

Worked Example: Simplify ln(x2ex) before differentiating.
Using log laws: ln(x2ex) = ln(x2) + ln(ex) = 2 ln x + x
Differentiating: d/dx[2 ln x + x] = 2/x + 1
This is far more efficient than applying the product rule directly inside a logarithm.

2. Differentiating Exponential Functions: d/dx[e^(f(x))] = f′(x)e^(f(x))

The derivative of the natural exponential function is its defining property: ex is its own derivative. When the exponent is itself a function of x — a composite form — the chain rule gives us the general result:

d/dx[ef(x)] = f′(x) · ef(x)

This result follows directly from the chain rule. Let u = f(x), so y = eu. Then dy/dx = (dy/du) · (du/dx) = eu · f′(x) = f′(x)ef(x). The original function is preserved and simply multiplied by the derivative of the exponent.

Immediate results:

  • d/dx[ex] = ex (the base case, where f(x) = x and f′(x) = 1)
  • d/dx[e3x] = 3e3x
  • d/dx[e−2x] = −2e−2x
  • d/dx[ex²] = 2xex²
  • d/dx[esin x] = cos x · esin x

Worked Example 1 — Simple chain rule:
Find dy/dx for y = e4x − 1.
Here f(x) = 4x − 1, so f′(x) = 4.
Therefore dy/dx = 4e4x − 1.

Worked Example 2 — Combining with the product rule:
Find d/dx[x² e3x].
Let u = x² and v = e3x.
Then u′ = 2x and v′ = 3e3x.
By the product rule: d/dx = u′v + uv′ = 2x · e3x + x² · 3e3x = e3x(2x + 3x²) = xe3x(2 + 3x).

Worked Example 3 — Quotient form:
Find d/dx[e2x / x].
Using the quotient rule with u = e2x, v = x:
d/dx = (2e2x · x − e2x · 1) / x² = e2x(2x − 1) / x².

A common Band A extension is to simplify the result by factoring out the exponential term, which is always positive and therefore never cancels a sign or creates a zero — useful when finding stationary points.

Sample exam question

A coastal seagrass biomass is modelled by B(x) = e^(3x^2). Using the chain rule, the derivative B'(x) is:

  • 6x e^(3x^2)
  • 3x e^(3x^2)
  • 6x e^(6x)
  • e^(3x^2)
Show the worked answer

Answer: A

Chain rule: B'(x) = e^(3x^2) x d/dx(3x^2) = e^(3x^2) x 6x = 6x e^(3x^2). Correct option: A.

All 20 practice exams

  1. Exam 1 — Unit 3 differentiation of exponential, logarithmic and trigonometric functions (chain, product, quotient rules); Applications of differentiation: instantaneous rate of change, kinematics, stationary points, second-derivative test, optimisation; Introduction to integration: antidifferentiation, reverse chain rule, definite integrals and signed area
  2. Exam 2 — Unit 3 Further Calculus: differentiation of exponential, logarithmic and trigonometric functions via chain, product and quotient rules; Applications of differentiation: rates of change, kinematics, stationary points, concavity, optimisation and curve interpretation; Introduction to integration: standard antiderivatives, reverse chain rule, definite integrals, area under a curve and between curves
  3. Exam 3 — Unit 3 differentiation: chain, product and quotient rules on exponential, logarithmic and polynomial functions; Unit 3 applications of differentiation: stationary points, concavity, points of inflection, kinematics and optimisation; Unit 3 introduction to integration: antidifferentiation, reverse chain rule, definite integrals and area under a curve
  4. Exam 4 — Unit 3 differentiation of exponential, logarithmic and trigonometric functions via chain, product and quotient rules; Applications of differentiation: rates of change, kinematics, stationary points, concavity and optimisation; Introduction to integration: standard antiderivatives, reverse chain rule, definite integrals and area
  5. Exam 5 — Unit 3 Further Calculus: chain/product/quotient differentiation of exp, log and trig functions; Applications of differentiation: kinematics, stationary points, concavity, points of inflection, optimisation; Antidifferentiation: standard forms, reverse chain rule, definite integrals and the constant of integration
  6. Exam 6 — Unit 3 differentiation: chain, product and quotient rules on exponential, logarithmic and trigonometric functions; Applications of differentiation: rates of change, kinematics, stationary points, curve sketching and optimisation; Introduction to integration: antidifferentiation, constant of integration from initial conditions, definite integrals and area
  7. Exam 7 — Differentiation of exponential, logarithmic and trigonometric functions (chain, product, quotient rules); Applications of differentiation: stationary points, kinematics, optimisation, curve sketching; Integration: antidifferentiation, definite integrals, signed area and area between curves
  8. Exam 8 — Unit 3 Further Calculus: differentiation of exponential, logarithmic and trigonometric functions via chain, product and quotient rules; Applications of differentiation: kinematics, stationary points, nature of stationary points, points of inflection, curve sketching and optimisation; Introduction to integration: antidifferentiation of standard forms, reverse chain rule, constant of integration from initial conditions, definite integrals and area
  9. Exam 9 — Differentiation of trigonometric, exponential and logarithmic functions via chain, product and quotient rules; Applications of differentiation: rates of change, kinematics, stationary points, concavity, inflection and optimisation; Antidifferentiation, reverse chain rule, definite integrals, area under and between curves, trapezoidal rule
  10. Exam 10 — Unit 3 differentiation: chain, product and quotient rules on exponential, logarithmic and trigonometric functions; Applications of differentiation: rates of change, kinematics, stationary points, concavity, points of inflection and optimisation; Integration: antidifferentiation of standard forms, reverse chain rule, definite integrals, area under curves and between curves
  11. Exam 11 — Differentiation of exponential, logarithmic and trigonometric functions (chain, product, quotient rules); Applications of differentiation: rates of change, kinematics, stationary points, optimisation; Integration: antidifferentiation, reverse chain rule, definite integrals, area under and between curves, trapezoidal rule
  12. Exam 12 — Differentiation: chain, product and quotient rules on exponential, logarithmic and trigonometric functions (Unit 3); Applications of differentiation: stationary points, inflections, concavity, kinematics and optimisation; Integration: standard antiderivatives, reverse chain rule, definite integrals, area under and between curves, trapezoidal rule
  13. Exam 13 — Unit 3 differentiation: chain, product, quotient rules on exponential, logarithmic and trigonometric functions; Unit 3 applications of differentiation: stationary points, nature, optimisation and kinematics; Unit 3 integration: antidifferentiation, constant of integration from initial conditions, definite integrals and area
  14. Exam 14 — Unit 3 Further Calculus: chain/product/quotient differentiation of exponential, logarithmic and trigonometric functions; Applications of differentiation: rates of change, kinematics, stationary points, optimisation; Introduction to integration: antidifferentiation, reverse chain rule, definite integrals and area
  15. Exam 15 — Unit 3 differentiation: chain, product, quotient rules on exponential, logarithmic and trigonometric functions; Applications of differentiation: kinematics, stationary points, concavity, points of inflection, curve sketching and optimisation; Introduction to integration: antidifferentiation of standard forms, reverse chain rule, definite integrals and area (with constant of integration)
  16. Exam 16 — Inferential statistics: sampling distribution of the sample proportion, margin of error, sample-size determination (n = z²p(1−p)/E²), and confidence intervals for a population proportion with precise written interpretation; Further calculus: chain, product and quotient rules on exponential, logarithmic and trigonometric functions; stationary points, concavity, points of inflection and optimisation; Integration: antidifferentiation of standard forms, reverse chain rule, definite integrals, signed area, area between curves and the trapezoidal rule
  17. Exam 17 — Unit 3 Further Calculus: differentiation of exponential, logarithmic and trigonometric functions via chain, product and quotient rules; Applications of differentiation: rates of change, kinematics, stationary points, concavity, inflection, curve sketching and optimisation; Integration: antidifferentiation of standard forms, reverse chain rule, definite integrals, signed area and area between curves
  18. Exam 18 — Unit 3 differentiation: chain, product and quotient rules for exponential, logarithmic and trigonometric functions; Applications of differentiation: kinematics, stationary points, concavity, optimisation and curve analysis; Integration: antidifferentiation of standard forms, reverse chain rule, definite integrals, area under curves and area between curves
  19. Exam 19 — Differentiation of exponential, logarithmic and trigonometric functions (chain, product, quotient rules); Applications of differentiation: rates of change, kinematics, stationary points, concavity, optimisation; Integration: antidifferentiation, reverse chain rule, definite integrals, signed area and area between curves
  20. Exam 20 — Differentiation (power, product, chain, quotient rules; logarithmic and exponential derivatives); Anti-differentiation and definite integrals (area under curves, area between curves, average value); Optimisation using calculus (open-top box, minimum-surface-area cylindrical can/container, maximum profit)

All 20 revision notes

  • Exponential and Logarithmic Derivatives
  • Derivatives of Trigonometric Functions
  • Chain, Product and Quotient Rules
  • Stationary Points, Concavity and Points of Inflection
  • Rates of Change and Kinematics
  • Optimisation Using Calculus
  • Antidifferentiation: Polynomials, Exponential, Log and Trig
  • Definite Integrals and Signed Area
  • Integration Applied to Motion
  • Binomial Distribution: Probability, Mean and Variance
  • Discrete Probability Distributions and Expected Value
  • Area Between Curves and Numerical Integration
  • Sine Rule, Cosine Rule and Triangle Area
  • Bearings, Elevation and 3D Geometry
  • Continuous Random Variables and PDFs
  • Normal Distribution: z-Scores, Probabilities and Inverse Normal
  • Sampling Distribution of the Sample Proportion
  • Confidence Intervals for a Population Proportion
  • Margin of Error and Required Sample Size
  • Integration of Derivative Functions and Area Interpretation