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# IB Maths .Basic differentiation

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### IB Maths .Basic differentiation

1. 1. Calculate the following limits:
2. 2. By the end of the lesson you will be able to: • Understand the definition of derivative. • Find derivatives from first principles. Differentiation • Calculate the gradient of a curve without drawing  the tangent line. • Relate derivatives and slopes of curves.
3. 3. Introduction to Differentiation http://www.youtube.com/watch?v=QC5ITOflh3k&feature=related
4. 4. Gradient of a line B A
5. 5. Gradient of a line B A y2 y1 x1 x2
6. 6. Gradient of a line B A y2 y1 x1 x2 The gradient of a line is constant.
7. 7. Gradient of a curve The gradient of a curve at a point  is the slope of  the tangent line at that point. http://www.math.umn.edu/%7Egarrett/qy/TraceTangent.html Tangent line to a curve.ggb
8. 8. Gradient of a curve The gradient of a curve varies at each point.
9. 9. We need to find a method to calculate the slope of  the tangent line at any point.
10. 10. x Let's consider a point P on the curve. P f(x) Coordinates of P :      ( x ,  f(x) ) f(x) Secant to tangent.ggb
11. 11. Coordinates of Q :      ( x+h ,  f(x+h) ) x Let's consider another point on the curve, Q. P f(x) Q x + h f(x)
12. 12. Coordinates of Q :      ( x+h ,  f(x+h) ) x Let's take another point on the curve , Q. P Q x + h f(x+h) f(x)
13. 13. x The line PQ is  secant  to the curve. If we find the  gradient of this line , it is not the tangent line but is a  starter. P f(x)Q x + h
14. 14. x P f(x)Q x + h f(x) f(x+h) m =
15. 15. x P f(x)Q x + h f(x) f(x+h) Gradient of secant line PQ : We can rewrite this gradient in a different way : Gradient of secant PQ :
16. 16. Q x + h Veamos que sucede si Q se acerca a P... P x Secant to tangent.ggb
17. 17. slope of the secant = x P f(x)Q x + h If Q gets closer to P...
18. 18. Then we need h to be as small as possible. h 0When the slope of the secant tends to be the slope of the tangent. This is the difference quotient , the definition of  the derivative.
19. 19. This  is called the  first derivative of function f with respect to x. Notation: f '(x)    or   y'     or
20. 20. Find the derivative of the function      f(x) = x2 Pp R S We will find the gradient of the tangent at any  point for the function  f(x) = x2
21. 21. f(x)  = x 2
22. 22. Pp R S m =4 m =-6 We found for f(x) = x2 which is a new function that gives the value of the gradient of the curve at each point. Para its derivative:
23. 23. Calculate the derivative of  f(x) = 3 x2  +1 To practice this topic on-line : http://archives.math.utk.edu/visual.calculus/2/definition.7/index.html
24. 24. The process of finding the derivative is called  differentiation.
25. 25. Book page 357 Ex 12B http://www.calculus­help.com/funstuff/phobe.html At the end of the lesson:
26. 26.   http://www.calculus­help.com/funstuff/phobe.html To revise this lesson at home:
27. 27. At the end of the lesson: http://animoto.com/play/hVIw4sOG1tEL3hYONxJyJQ
28. 28. Attachments Tangent line to a curve.ggb Secant to tangent.ggb