1st Year Math Chapter 13 Exercise 13.2 Notes: Theorems (Rules) of Differentiation (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 13.2 introduces the standard differentiation rules — shortcuts that let you find derivatives quickly without repeating the first-principles limit process from Exercise 13.1 every time.
What Does This Exercise Cover?
This exercise teaches the power rule, the constant multiple and sum/difference rules, and the product and quotient rules, giving you the practical toolkit used to differentiate almost any combination of functions at this level.
Key Concepts
1. Derivative of a Constant
For any constant c, d/dx(c) = 0, since a constant function never changes and so has a zero rate of change everywhere.
2. The Power Rule
For any real number n, d/dx(xⁿ) = n·xⁿ⁻¹. This single rule covers polynomials, roots (written as fractional powers), and reciprocals (written as negative powers).
3. Constant Multiple, Sum, and Difference Rules
A constant factor can be pulled outside the derivative: d/dx[c·f(x)] = c·f′(x). Sums and differences differentiate term by term: d/dx[f(x) ± g(x)] = f′(x) ± g′(x).
4. The Product Rule
For two functions multiplied together: d/dx[f(x)·g(x)] = f′(x)·g(x) + f(x)·g′(x).
5. The Quotient Rule
For one function divided by another (with g(x) ≠ 0): d/dx[f(x)/g(x)] = [f′(x)·g(x) − f(x)·g′(x)] / [g(x)]².
Basic Differentiation Rules at a Glance
| Rule | Formula |
| Constant | d/dx(c) = 0 |
| Power | d/dx(xⁿ) = n·xⁿ⁻¹ |
| Constant multiple | d/dx[c·f(x)] = c·f′(x) |
| Sum/Difference | d/dx[f ± g] = f′ ± g′ |
| Product | d/dx[f·g] = f′g + fg′ |
| Quotient | d/dx[f/g] = (f′g − fg′) / g² |
Solved Examples
Example 1: Differentiate y = x⁴ − 3x² + 2x with respect to x.
Apply the power rule to each term: d/dx(x⁴) = 4x³, d/dx(3x²) = 6x, d/dx(2x) = 2.
Answer: dy/dx = 4x³ − 6x + 2
Example 2: Differentiate y = (x² + 1)(x − 3) using the product rule.
Let f = x² + 1, g = x − 3, so f′ = 2x, g′ = 1.
dy/dx = f′g + fg′ = 2x(x−3) + (x²+1)(1) = 2x² − 6x + x² + 1.
Answer: dy/dx = 3x² − 6x + 1
Example 3: Differentiate y = (2x + 3)/(x − 1) using the quotient rule.
Let f = 2x + 3, g = x − 1, so f′ = 2, g′ = 1.
dy/dx = [2(x−1) − (2x+3)(1)] / (x−1)² = [2x − 2 − 2x − 3] / (x−1)².
Answer: dy/dx = −5 / (x−1)²
Sample MCQs
1. The derivative of a constant is:
a) 1 b) 0 c) The constant itself d) Undefined
Answer: b) 0
2. The power rule states that d/dx(xⁿ) equals:
a) xⁿ⁻¹ b) n·xⁿ⁻¹ c) n·xⁿ d) xⁿ⁺¹
Answer: b) n·xⁿ⁻¹
3. The product rule for differentiating f(x)·g(x) is:
a) f′(x)·g′(x) b) f′(x)g(x) + f(x)g′(x) c) f′(x) + g′(x) d) f′(x)g(x) − f(x)g′(x)
Answer: b) f′(x)g(x) + f(x)g′(x)
4. The quotient rule requires that:
a) f(x) = 0 b) g(x) ≠ 0 c) f(x) = g(x) d) Both f and g are constants
Answer: b) g(x) ≠ 0
5. The derivative of y = 5x³ is:
a) 5x² b) 15x² c) 3x² d) 15x
Answer: b) 15x²
Important Short Questions
- State the power rule for differentiation.
- State the sum and difference rule for differentiation.
- Differentiate y = 3x² − 4x + 7 with respect to x.
- State the product rule for differentiation.
- State the quotient rule for differentiation, including any required condition.
Important Long Questions
- Differentiate y = 2x⁴ − 5x³ + x − 9 with respect to x, showing each term’s derivative.
- Differentiate y = (x² + 1)(x − 3) using the product rule.
- Differentiate y = (2x + 3)/(x − 1) using the quotient rule.
- Differentiate y = 3x⁻² + 2√x with respect to x, using the power rule for negative and fractional exponents.
How to Approach This Exercise Effectively
- Rewrite roots and reciprocals as powers of x (like √x = x^(1/2), 1/x² = x⁻²) before applying the power rule — this avoids confusion.
- For products of two simple expressions, decide whether expanding first is faster than using the product rule — both approaches give the same answer.
- In the quotient rule, keep track of which function is f and which is g consistently; swapping them changes the sign of your answer.
- Differentiate term-by-term for sums and differences — there’s no need to treat the whole expression as one block.
- Double-check quotient rule answers by simplifying the numerator fully before finalizing, since sign errors are the most common mistake there.
FAQs
Q: Why is the derivative of a constant always zero?
A: A constant function’s graph is a horizontal line, which has zero slope everywhere — since the derivative measures slope, it must be zero.
Q: When should the product rule be used instead of just expanding the expression first?
A: For simple polynomial products, expanding first is often quicker; the product rule becomes essential when the functions involved can’t easily be multiplied out, like more complex expressions.
Q: Can the quotient rule be used when the denominator is a constant?
A: Yes, though it’s usually faster to instead treat a constant denominator as a constant multiple (dividing by a constant is the same as multiplying by its reciprocal) rather than applying the full quotient rule.
Q: How is the power rule extended to negative and fractional exponents?
A: The rule d/dx(xⁿ) = n·xⁿ⁻¹ holds for any real number n, so negative exponents (like x⁻¹) and fractional exponents (like x^(1/2) for √x) are differentiated the exact same way as positive integer powers.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 13: Differentiation, Exercise 13.2.