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Www.le.ac.uk Differentiation Department of Mathematics University of Leicester.

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Presentation on theme: "Www.le.ac.uk Differentiation Department of Mathematics University of Leicester."— Presentation transcript:

1 Differentiation Department of Mathematics University of Leicester

2 Content Differentiation of functionsIntroduction

3 Differentiation is the process of finding the rate of change of some quantity (eg. a line), at a general point x. The rate of change at x is equal to the gradient of the tangent at x. We can approximate the gradient of the tangent using a straight line joining 2 points on the graph… Next FunctionsIntroduction

4 Choose h =. from to : The straight line has a gradient of : Next FunctionsIntroduction Gradient of tangent at is (Tangent Line at 0.5)

5 Gradients The gradient of the line from to is. ie. its the difference between the 2 points. As h gets smaller the line gets closer to the tangent, so we let h tend to 0. We get: Next FunctionsIntroduction

6 Differentiation When you differentiate f(x), you find This is called the derivative, and is written as or, or (for example). Each function has its own derivative... FunctionsIntroduction Next

7 Summary Click on the functions to see how they are derived. FunctionsIntroduction Next

8 Differentiating a constant Next FunctionsIntroduction Back to summary

9 Differentiating : This is using the binomial expansion Next FunctionsIntroduction Back to summary The Binomial expansion gives a general formula for (x+y) n. It says:

10 Next FunctionsIntroduction Back to summary All these terms contain h, so disappear when we take the limit as h 0

11 Differentiating : Next FunctionsIntroduction Back to summary The Maclaurins Series gives an expansion for e x. It says:

12 Next FunctionsIntroduction Back to summary This is using the Maclaurins Series for e h. The Maclaurins Series gives an expansion for e x. It says:

13 Differentiating : This is using the Trigonometric Identity for sin(a+b) NextBack to summary FunctionsIntroduction The Trig Identity says:

14 This is using the Maclaurins Series for sin(x) and cos(x) NextBack to summary FunctionsIntroduction So The Maclaurins Series gives expansions for sinx and cosx It says:

15 Differentiating : This is using the Trigonometric Identity for cos(a+b) NextBack to summary FunctionsIntroduction The Trig Identity says:

16 This is using the Maclaurins Series for sin(x) and cos(x) NextBack to summary FunctionsIntroduction So The Maclaurins Series gives expansions for sinx and cosx It says:

17 Differentiating : Next FunctionsIntroduction Back to summary `

18 This is using the Macluarins Series for ln(a+1) Next FunctionsIntroduction Back to summary Because after you divide by h, all the other terms have h in them so disappear as h 0. The Maclaurins Series gives an expansion for ln(a + 1). It says:

19 Next FunctionsIntroduction Back to summary Differentiating :

20 Next FunctionsIntroduction Back to summary Differentiating :

21 Questions differentiates to: FunctionsIntroduction

22 Questions differentiates to: FunctionsIntroduction

23 Questions differentiates to: FunctionsIntroduction

24 Questions differentiates to: FunctionsIntroduction

25 Questions differentiates to: FunctionsIntroduction

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27 Questions differentiates to: FunctionsIntroduction

28 Questions differentiates to: FunctionsIntroduction

29 Conclusion Differentiation is the process of finding a general expression for the rate of change of a function. It is defined as Differentiation is a process of subtraction. Using this official definition, we can derive rules for differentiating any function. Next FunctionsIntroduction

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