1. www.le.ac.uk 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 : 0.5 1 1.522.5 Next FunctionsIntroduction Gradient of tangent at is 0.4. 0 (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 :

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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