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

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Content Product Rule Quotient Rule Chain RuleInversion RuleIntroduction

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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

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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

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Let. Then: Go back to Product Rule

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Product rule example Find. Next Product Quotient ChainInversion Intro

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Take: Next Differentiate these: Product Quotient ChainInversion Intro

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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

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Let. Then: Go back to Quotient Rule

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Quotient rule example Find. Next Product Quotient ChainInversion Intro

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

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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

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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

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We have. Go back to Chain Rule

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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

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

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

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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

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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

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

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

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

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

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

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Differentiate Product Quotient ChainInversion Intro

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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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