Differential and Integral Calculus for CA Foundation: Study Order, Formulas and MCQ Shortcuts

What this chapter covers

Differential and Integral Calculus: topics in the order to study them

1. 1Limits and Continuity BasicsThe derivative is defined as a limit, so you need this idea first, and it is also tested directly.
2. 2Differentiation Rules and Standard FormulasThis is the core skill of the chapter. Every later topic depends on it.
3. 3Higher Order DerivativesIt only repeats differentiation, and you need the second derivative for maxima and minima.
4. 4Maxima and MinimaIt uses the first and second derivative together to find the highest and lowest values.
5. 5Applications of Differentiation: Cost, Revenue and Marginal AnalysisIt applies maxima and minima to business questions such as profit and average cost.
6. 6Integration Basics and Standard FormulasIntegration is the reverse of differentiation, so it comes once your derivatives are secure.
7. 7Integration by Parts and Partial FractionsThese are techniques for harder integrals and need the basic formulas to be automatic.
8. 8Definite Integrals and Their ApplicationsIt uses all earlier integration skills and adds limits, areas and business totals.

How to prepare Differential and Integral Calculus

1. Write the standard derivatives and integrals on one page. Learn them until you can write them from memory in two minutes.
2. Study limits briefly. Focus on factorising and cancelling for 0/0 cases, and on the condition for continuity.
3. Practise differentiation until the product, quotient and chain rules feel automatic. Do 10 to 15 mixed problems per session.
4. For maxima and minima, follow a fixed routine: set f'(x) = 0, solve, check f''(x), then find the value. Use it for cost and revenue problems too.
5. Learn integration as reverse differentiation. After each answer, differentiate it mentally to check, and never forget the constant of integration in indefinite integrals.
6. Practise definite integrals by substituting the upper limit minus the lower limit. Then solve timed MCQ sets and review every wrong answer.
7. In the last week, redo only your error list and the formula page. Skip any long question in the exam and return to it if time remains.

Common mistakes in Differential and Integral Calculus

• Applying n·xⁿ⁻¹ to eˣ or to aˣ, giving x·eˣ⁻¹.

  Fix: Ask first whether the variable is in the base or in the exponent. If x is in the exponent, use the eˣ or aˣ formula.

• Forgetting the chain rule, for example differentiating (3x + 2)⁵ as 5(3x + 2)⁴ only.

  Fix: Always multiply by the derivative of the inside. Here the answer is 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴.

• Declaring a maximum or minimum from f'(x) = 0 alone.

  Fix: Find f''(x) at each point. Negative means maximum, positive means minimum. If it is zero, the test fails and you need another check.

• Dropping the constant C in indefinite integrals, or adding it in definite integrals.

  Fix: Indefinite integrals need + C. Definite integrals have limits, and C cancels, so do not write it.

• Using ∫ (1/x) dx = x⁰ ÷ 0 through the power formula, or treating n = −1 like other powers.

  Fix: Remember n = −1 is the exception and the answer is log x + C.

• Mixing up total cost with average cost or marginal cost in business questions.

  Fix: Write the definition first: marginal = derivative of total; average = total ÷ x. Then differentiate or divide accordingly.

Last-day revision: Differential and Integral Calculus

• ✓d/dx (xⁿ) = n·xⁿ⁻¹ for any real n.
• ✓d/dx (eˣ) = eˣ and d/dx (log x) = 1/x for x > 0 (log means natural log).
• ✓d/dx (aˣ) = aˣ · log a for a > 0.
• ✓Product rule: d(uv)/dx = u·dv/dx + v·du/dx.
• ✓Quotient rule: d(u/v)/dx = (v·du/dx − u·dv/dx) ÷ v², for v ≠ 0.
• ✓Chain rule: d/dx f(g(x)) = f'(g(x)) · g'(x).
• ✓For maxima or minima, set f'(x) = 0. If f''(x) < 0 it is a maximum; if f''(x) > 0 it is a minimum.
• ✓Marginal cost = d(Total Cost)/dx and marginal revenue = d(Total Revenue)/dx.
• ✓∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C, for n ≠ −1; ∫ (1/x) dx = log x + C for x > 0.
• ✓∫ eˣ dx = eˣ + C, and ∫ aˣ dx = aˣ ÷ log a + C for a > 0, a ≠ 1.
• ✓Integration by parts: ∫ u·v dx = u∫v dx − ∫ (du/dx · ∫v dx) dx.
• ✓Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a); the constant C cancels out.

Differential and Integral Calculus practice questions

• Evaluate ∫(8x³ − 6x + 5) dx.
• A manufacturing company's profit function is given by P(x) = -2x² + 80x - 300, where x is the number of units produced (in hundreds). At wha…
• A firm's total cost function is C(x) = x² + 3600, where x is the number of units produced. The output at which the average cost per unit is …
• The value of the definite integral of (3x² + 1) with respect to x from x = 0 to x = 2 is:
• Evaluate the definite integral ∫₀² (6x² + 4x) dx.
• The profit function of a company is P(x) = −x² + 40x − 100 (in ₹), where x is the output in units. The maximum profit is:
• If f(x) = 3x² + 5x + 2, find the derivative f'(x).
• The demand law for a product is p = 49 − x², where p is the price in ₹ and x is the quantity demanded. If the market price is ₹40, the consu…