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Mastering Differentiation: Gradients, Rates of Change & Implicit Differentiation

Explore differentiation principles, gradients, rates of change, and implicit differentiation techniques. Practice problems included. Learn how to find derivatives using first principles and handle implicit functions effectively.

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Mastering Differentiation: Gradients, Rates of Change & Implicit Differentiation

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  1. Differentiation First Principles & Implicit Differentiation

  2. First Principles • What is differentiation? • Gradients • Rates of Change • First Principles

  3. Gradient Not that great.

  4. Gradient Better.

  5. Gradient Even better again.

  6. Gradient • A shorter ‘’ is a better estimate of gradient • What is the best ‘’? • 0, of course! • However, that’s a problem – we can’t divide by zero! • Time to cheat

  7. First Principles • We start with the curve , and our of • The is the change in over the duration of the • This means we have • To make , we write (using limits)

  8. Example 1 We know the answer will be • Find the derivative of using first principles.

  9. Example 2 We need the identity • Find the derivative of using first principles.

  10. Practice • Delta Workbook • Exercise 5.7, page 77 • Workbook • Pages 15-20

  11. Implicit Differentiation • We know that a function like becomes • So the becomes when we differentiate • If the is in the middle of the equation, is this still true? • Yes!

  12. Example 1 This equation is the same as • Differentiate implicitly:

  13. Implicit Differentiation • What if we have more than one , e.g. ? • We can use the chain rule!

  14. Example 2 This is really horrible to rearrange – use implicit differentiation! • Differentiate • There is still a in the answer – this is allowed!

  15. Practice • Delta Workbook • Exercise 13.1, page 139 • Workbook • Pages 27-31, 32-35 (harder)

  16. Do Now • Any Questions? • Delta Workbook • Exercises 5.7, 13.1 • Workbook • Pages 15-20, 27-35

  17. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Aaron Stockdill 2016

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