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Www.le.ac.uk Differentiation – Product, Quotient and Chain Rules Department of Mathematics University of Leicester.

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1 www.le.ac.uk Differentiation – Product, Quotient and Chain Rules Department of Mathematics University of Leicester

2 Content Product Rule Quotient Rule Chain RuleInversion RuleIntroduction

3 Previously, we differentiated simple functions using the definition: Now, we introduce some rules that allow us to differentiate any complex function just by remembering the derivatives of the simple functions… Next Product Quotient ChainInversion Intro

4 The product rule is used for functions like: where and are two functions. The product rule says: Differentiate the 1 st term and times it by the 2 nd, then differentiate the 2 nd term and times it by the 1 st. Product rule Product Quotient ChainInversion Intro Click here for a proof Next

5 Let. Then: Go back to Product Rule

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7 Product rule example Find. Next Product Quotient ChainInversion Intro

8 Take: Next Differentiate these: Product Quotient ChainInversion Intro

9 The quotient rule is used for functions like: where and are two functions. The quotient rule says: This time, it’s a subtraction, and then you divide by. Quotient rule Next Product Quotient ChainInversion Intro Click here for a proof

10 Let. Then: Go back to Quotient Rule

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13 Quotient rule example Find. Next Product Quotient ChainInversion Intro

14 Take: (give your answers as decimals) Next Differentiate these: Product Quotient ChainInversion Intro

15 The chain rule is used for functions,, which have one expression inside another expression. Let be the inside part, so that now is just a function of. Then the chain rule says: Chain rule Next, which has inside., then Product Quotient ChainInversion Intro Click here for a proof

16 Instead of writing: We write: The best way to prove the chain rule is to write the definition of derivative in a different way: If we put, we see that these two definitions are the same. Go back to Chain Rule

17 We have. Go back to Chain Rule

18 u(x) is just u, and u(a) is just a number, so we can call it b. Then the first term matches the definition of. Go back to Chain Rule

19 Chain rule example Find. Next, so Product Quotient ChainInversion Intro

20 Let: Let:, and inside that, let:Let: differentiates to Next True or False? differentiates to Product Quotient ChainInversion Intro

21 Inversion Rule If you have a function that is written in terms of y, eg. Then you can use this fact: So if, then. Next Product Quotient ChainInversion Intro Click here for a proof

22 First note that, because we’re differentiating the function. Then: This is a function, so we can divide by it… We get:. by the chain rule, Go back to Inversion Rule

23 Inversion Rule Example A curve has an equation. Find when., therefore Then when, Next Product Quotient ChainInversion Intro

24 Note, it is NOT TRUE that Next Product Quotient ChainInversion Intro

25 Next Find at the specified values of y: Product Quotient ChainInversion Intro

26 More complicated example Find. Quotient rule: Chain rule on : – is ‘inside’, so let –Then, so Next Product Quotient ChainInversion Intro

27 Chain rule on also gives. Then quotient rule gives: Next Product Quotient ChainInversion Intro

28 Differentiate Product Quotient ChainInversion Intro

29 Conclusion We can differentiate simple functions using the definition: We have found rules for differentiating products, quotients, compositions and functions written in terms of x. Using these two things we can now differentiate ANY function. Next Product Quotient ChainInversion Intro

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