Overview
This lecture covers 100 key objective questions and techniques related to derivatives and differentiation, focusing on commonly tested formats in board exams with step-by-step approaches and important formulas.
Types of Differentiation Problems
- Differentiation of polynomial, exponential, trigonometric, and logarithmic functions is emphasized.
- Common forms include chain rule, product rule, quotient rule, and parametric differentiation.
- Board exams frequently test derivative shortcuts and standard forms.
Important Formulas & Techniques
- The derivative of ( x^n ) is ( n x^{n-1} ).
- The derivative of ( e^{ax} ) is ( a e^{ax} ).
- The derivative of ( a^x ) is ( a^x \ln a ).
- The derivative of ( \sin(ax) ) is ( a \cos(ax) ); for ( \cos(ax) ), it's ( -a \sin(ax) ).
- The derivative of ( \ln x ) is ( 1/x ).
- For composite functions ( f(g(x)) ), use the chain rule: ( f'(g(x)) \cdot g'(x) ).
- For ( y = a^{f(x)} ), ( \frac{dy}{dx} = a^{f(x)} \ln a \cdot f'(x) ).
- The product rule: ( (uv)' = u'v + uv' ).
- The quotient rule: ( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} ).
Differentiation of Special Functions
- Parametric derivatives: ( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} ).
- Inverse trigonometric derivatives, e.g., ( \frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1-x^2}} ).
- Differentiation under radicals: ( \frac{d}{dx} \sqrt{f(x)} = \frac{f'(x)}{2 \sqrt{f(x)}} ).
- If a function repeats, as in infinite surds or perpetuities, reduce using pattern recognition.
Board-Exam Oriented Tips
- Focus on direct formulas for objective questions to save time.
- Practice recognition of repeated forms in options to avoid common traps.
- Memorize standard derivatives for all basic algebraic, trigonometric, exponential, and logarithmic functions.
Key Terms & Definitions
- Derivative — The instantaneous rate of change of a function with respect to its variable.
- Chain Rule — Method for differentiating composite functions.
- Product Rule — Technique for differentiating products of two functions.
- Quotient Rule — Technique for differentiating ratios of two functions.
- Parametric Differentiation — Derivative when variables are given in terms of a parameter.
- Inverse Trigonometric Functions — Functions like ( \sin^{-1}(x) ), ( \cos^{-1}(x) ), etc., with special derivative rules.
Action Items / Next Steps
- Review all 100 objective-type differentiation problems discussed.
- Practice similar derivative questions from board-level sample papers.
- Memorize and revise the key formulas and differentiation shortcuts.
- Watch or review the referenced previous class/videos for foundational concepts.