Exercise 2.2 Class 12 Math Solutions - Unit 2 Further Differentiation, Punjab Board
Step-by-step solutions to Exercise 2.2, Unit 2: Further Differentiation, for Class 12 Punjab Board students. Covers differentiation of natural and general exponential functions, natural and base-a logarithmic functions, and logarithmic differentiation for complex product/quotient expressions and variable exponents [f(x)]^g(x), worked out question by question.
Exercise Overview
Exercise 2.2 of Class 12 Mathematics addresses one of the most essential topics in differential calculus: the differentiation of exponential and logarithmic functions under the Punjab Board PCTB / PECTAA curriculum. The exercise establishes the fundamental derivative rules for natural exponential functions (d/dx (eˣ) = eˣ), general base exponentials (d/dx (aˣ) = aˣ·ln(a)), natural logarithms (d/dx (ln x) = 1/x), and general logarithms (d/dx (logₐ x) = 1/(x · ln a)).
A major emphasis in this exercise is placed on composite chain rule applications and the powerful technique of Logarithmic Differentiation. When dealing with complex expressions involving multiple multiplied, divided, or radical factors, or functions where both base and exponent vary with x (such as y = [f(x)]^g(x)), taking the natural logarithm of both sides simplifies the expression into manageable sums and differences before differentiating implicitly.
For Punjab Board HSSC Part 2 examinations across all nine educational boards (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 2.2 is a vital source of board exam short questions, derivative evaluations, and multi-step long problems. These step-by-step solutions provide rigorous algebraic manipulation and clear step breakdowns for every question.