CA Foundation Course

Differential and Integral Calculus (basics) for CA Foundation

Basics of differentiation and integration: standard derivatives and integrals, rules, and simple applications like marginal cost and maxima and minima.

Quantitative AptitudePaper 3, Business Mathematics12 MCQs2 lessons2 flashcards3 memory tricks

Practise Differential and Integral Calculus (basics) free

What you'll learn

  1. Differentiation
    Differentiate standard functions and apply the rules.
  2. Integration
    Integrate standard functions and evaluate definite integrals.

Key formulas

xⁿnxⁿ⁻¹
eˣeˣ
aˣaˣ log a
log x1/x
constant0
Product(uv)′ = uv′ + vu′
Quotient(u/v)′ = (vu′ − uv′)/v²
Chaindy/dx = (dy/du)(du/dx)

Concepts explained

Standard derivatives and rules

Differentiation measures the rate of change. Learn the standard derivatives and the product, quotient and chain rules.

What it means

  • d/dx xⁿ = nxⁿ⁻¹; constant → 0.
  • d/dx eˣ = eˣ; d/dx aˣ = aˣ log a; d/dx log x = 1/x.
  • Product: uv′ + vu′; quotient: (vu′ − uv′)/v².
  • Chain rule: dy/dx = dy/du × du/dx.
Example 1: Power
  • y = x⁵

Answer: 5x⁴

Example 2: At a point
  • y = x³ at x = 2

Answer: 12

Related terms

  • Second derivative: rate of change of the slope
  • Slope of tangent: value of dy/dx at a point

Exam tip: Differentiate first, substitute after.

Marginal cost and maxima/minima

In business maths, derivatives give marginal cost and revenue, and help find maximum profit or minimum cost.

What it means

  • Marginal cost = dC/dx; marginal revenue = dR/dx.
  • Profit is maximum where MR = MC (dP/dx = 0) and d²P/dx² < 0.
  • Minimum: dy/dx = 0, d²y/dx² > 0.
  • Average cost = total cost ÷ output.
Example 1: Marginal cost
  • C = 100 + 5x + x², x = 10

Answer: 25

Example 2: Max profit
  • P = 40x − x² − 100

Answer: x = 20, P = 300

Related terms

  • Elasticity: uses derivatives of demand
  • Break-even: revenue = cost

Exam tip: Check the second derivative's sign to tell a maximum from a minimum.

Integration basics

Integration reverses differentiation. Indefinite integrals carry a constant C; definite integrals give a number.

What it means

  • ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1).
  • ∫ 1/x dx = log |x| + C; ∫ eˣ dx = eˣ + C.
  • ∫ₐᵇ f(x) dx = F(b) − F(a).
  • Total cost = ∫ MC dx + fixed cost.
Example 1: Indefinite
  • ∫ x² dx

Answer: x³/3 + C

Example 2: Definite
  • ∫₀² 2x dx

Answer: 4

Related terms

  • Antiderivative: a function whose derivative is the integrand
  • Area under a curve: given by a definite integral

Exam tip: Check an integral by differentiating your answer.

Memory tricks

Power rulesDifferentiate: power comes down and drops one. Integrate: power goes up one and goes under.

d/dx xⁿ = nxⁿ⁻¹; ∫ xⁿ dx = xⁿ⁺¹/(n + 1).

Hilltop and valleyHilltop: flat and bending down (y″ < 0). Valley: flat and bending up (y″ > 0).

Maximum: y′ = 0, y″ < 0; minimum: y′ = 0, y″ > 0.

MarginalMarginal means 'one more': the derivative of the total.

MC = dTC/dx; MR = dTR/dx.

Try 3 CA Foundation-style questions

Q1. A firm's total cost is C = 100 + 5x + x². The marginal cost when x = 10 is

  1. 250
  2. 25
  3. 15
  4. 205
Show answer

Answer: (2) 25

Marginal cost = dC/dx = 5 + 2x = 5 + 20 = 25.

Q2. ∫ x² dx equals

  1. 2x + C
  2. x³ + C
  3. x³/3 + C
  4. 3x³ + C
Show answer

Answer: (3) x³/3 + C

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, so ∫ x² dx = x³/3 + C.

Q3. ∫ (1/x) dx equals

  1. x⁰/0 + C
  2. log x + C
  3. −1/x² + C
  4. x + C
Show answer

Answer: (2) log x + C

The power rule fails for n = −1; instead ∫ (1/x) dx = log |x| + C.

Practise all 12 questions Free · explanations for every option · see where you go wrong

More Quantitative Aptitude chapters (Paper 3, Business Mathematics)

Ratio, Proportion, Indices and Logarithms · Equations · Linear Inequalities · Mathematics of Finance · Permutations and Combinations · Sequence and Series · Sets, Relations and Functions; Limits and Continuity