Exercise 13.1 Solutions - Class 11 Mathematics Unit 13 Differentiation

Step-by-step solved problems, first-principle derivations, and theorems for Exercise 13.1, Unit 13: Differentiation.Comprehensive solutions covering the Ab-Initio method, limit of difference quotient, power rule derivation, and differentiability vs continuity for Punjab Board Class 11 (PCTB).

This page provides complete, step-by-step solved solutions for Exercise 13.1 of Unit 13: Differentiation for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All problems on finding derivatives by the definition (first principles / ab-initio method), interpreting derivatives as slopes of tangents and instantaneous rates of change, deriving the general power formula d/dx(xⁿ) = n xⁿ⁻¹ via binomial expansion, and proving that differentiability implies continuity are fully worked out for HSSC Part 1 exam prep.
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Exercise Overview

Exercise 13.1 opens Unit 13: Differentiation in Class 11 Mathematics, introducing the foundational concept of the derivative as the central tool of differential calculus.

The exercise establishes both the geometric perspective (where the derivative represents the limiting slope of secant lines transitioning into a tangent line to a curve) and the physical perspective (instantaneous rate of change of position or velocity). Students master the systematic 4-step Ab-Initio (first principles) technique to compute f'(x) = lim (δx → 0) [f(x + δx) - f(x)] / δx for algebraic expressions, rational powers, and reciprocal functions. It also proves the fundamental power formula d/dx(xⁿ) = n xⁿ⁻¹ for integer exponents using the Binomial Theorem and investigates the theorem stating that differentiability at a point implies continuity.

In Punjab Board HSSC Part 1 examinations, Exercise 13.1 is a fundamental topic, frequently providing Section B short questions and Section C theorem-based long questions across Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, and other BISE boards. Mastering the first-principles method provides deep mathematical grounding essential for calculus rules, physics mechanics, and higher applied engineering.

FAQs

Are these Exercise 13.1 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly follow the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) Class 11 Mathematics syllabus for Unit 13 (Differentiation) for the 2026-2027 academic session.
What are the key concepts covered in Exercise 13.1 Class 11 Mathematics?
Exercise 13.1 covers the definition of derivative as the limit of a difference quotient, slope of tangent lines, rate of change, the four-step ab-initio method, derivation of xⁿ, and the link between differentiability and continuity.
What is the definition of a derivative (first principles / ab-initio method)?
The derivative of f(x) is defined as f'(x) = lim (δx → 0) [f(x + δx) - f(x)] / δx, representing the instantaneous rate of change of f(x) with respect to x.
What are the four steps of the ab-initio method?
Step 1: Write y + δy = f(x + δx). Step 2: Subtract y to find δy = f(x + δx) - f(x). Step 3: Divide both sides by δx. Step 4: Evaluate lim (δx → 0) (δy/δx) to get dy/dx.
How is the power rule d/dx(xⁿ) = n xⁿ⁻¹ proved using first principles?
Substitute f(x) = xⁿ into the definition, expand (x + δx)ⁿ using the Binomial Theorem, cancel leading xⁿ terms, factor out δx, and let δx → 0 to obtain n xⁿ⁻¹.
What is the relationship between differentiability and continuity?
If a function is differentiable at a point, it is guaranteed to be continuous at that point. However, continuity does not imply differentiability (e.g., sharp corners like f(x) = |x| at x = 0).
Can students read these Exercise 13.1 solutions online?
Yes, students can read all Exercise 13.1 step-by-step derivations and first-principles proofs directly online on MaryamNotes.pk on any web browser without downloading.
Are these Class 11 Exercise 13.1 notes completely free to access?
Yes, all solved exercise notes, derivative proofs, and calculus resources on MaryamNotes.pk are 100% free and open to all students and teachers across Pakistan.