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Mastering Differentiation: Fundamental Rules, Product, Quotient, and Beyond

This session focuses on essential differentiation techniques in calculus, including fundamental rules, product and quotient rules, and differentiation of functions of a function (chain rule). We also cover trigonometric substitutions and implicit differentiation, providing step-by-step solutions for each method. Through examples and class exercises, students will learn how to differentiate complex functions effectively. By the end of the session, students will be equipped to tackle a variety of differentiation problems with confidence and clarity.

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Mastering Differentiation: Fundamental Rules, Product, Quotient, and Beyond

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Presentation Transcript

  1. Mathematics

  2. Session Differentiation - 2

  3. Session Objectives • Fundamental Rules, Product Rule and Quotient Rule • Differentiation of Function of a Function • Differentiation by Trigonometric Substitutions • Differentiation of Implicit Functions • Class Exercise

  4. Fundamental Rules

  5. Differentiate the following: Example-1

  6. Solution Differentiating y with respect to x, we get

  7. are differentiable functions, then If and Product Rule

  8. Differentiate: w.r.t. x. Example-2 Solution Let y =x2sinxlogx Differentiating w.r.t. x, we get

  9. , then are differentiable functions and If and Quotient Rule

  10. Differentiate: w.r.t. x. Example-3 Solution: Differentiating w.r.t. x, we get

  11. , then and Note: If Differentiation of Function of a Function If ƒ(x) and g(x) are differentiable functions, then ƒog is also differentiable (Chain Rule)

  12. Differentiate w. r. t. x. Example-4 Solution: Differentiating y w.r.t. x, we get

  13. Example-5

  14. Continued

  15. Trigonometric Substitutions

  16. Example-6 Solution: Putting x2 = cos2q

  17. Continued Differentiating y w.r.t. x, we get

  18. Example-7

  19. Continued

  20. Differentiation of Implicit Functions y is not expressible directly in terms of x

  21. Example-8 Solution: We have xy3 – yx3 = x

  22. Solution Cont.

  23. Thank you

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