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

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Content Differentiation of functionsIntroduction

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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

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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)

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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

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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

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

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Differentiating a constant Next FunctionsIntroduction Back to summary

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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:

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

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

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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:

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

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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:

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

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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:

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Differentiating : Next FunctionsIntroduction Back to summary `

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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:

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Next FunctionsIntroduction Back to summary Differentiating :

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Next FunctionsIntroduction Back to summary Differentiating :

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Questions differentiates to: FunctionsIntroduction

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Questions differentiates to: FunctionsIntroduction

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Questions differentiates to: FunctionsIntroduction

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Questions differentiates to: FunctionsIntroduction

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Questions differentiates to: FunctionsIntroduction

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Questions differentiates to: FunctionsIntroduction

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Questions differentiates to: FunctionsIntroduction

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Questions differentiates to: FunctionsIntroduction

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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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Pg. 407/423 Homework Pg. 407#33 Pg. 423 #16 – 18 all #19 Ѳ = kπ#21t = 0.52 + 2kπ, 2.62 + 2kπ #23 x = π/2 + 2kπ#25x = π/6 + 2kπ, 5π/6 + 2kπ #27 x = ±1.05.

Pg. 407/423 Homework Pg. 407#33 Pg. 423 #16 – 18 all #19 Ѳ = kπ#21t = 0.52 + 2kπ, 2.62 + 2kπ #23 x = π/2 + 2kπ#25x = π/6 + 2kπ, 5π/6 + 2kπ #27 x = ±1.05.

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