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**Differentiation – Product, Quotient and Chain Rules**

Department of Mathematics University of Leicester

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**Content Introduction Product Rule Quotient Rule Chain Rule**

Inversion Rule

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Intro Product Quotient Chain Inversion Introduction 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

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**Product rule The product rule is used for functions like:**

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

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Go back to Product Rule Let Then:

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Go back to Product Rule

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**Product rule example Find . Intro Product Quotient Chain Inversion**

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**Differentiate these: Intro Product Quotient Chain Inversion Take:**

Next

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**Quotient rule The quotient rule is used for functions like:**

Intro Product Quotient Chain Inversion Quotient rule 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 . Click here for a proof Next

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**Go back to Quotient Rule**

Let Then:

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**Go back to Quotient Rule**

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**Go back to Quotient Rule**

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**Quotient rule example Find . Intro Product Quotient Chain Inversion**

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**(give your answers as decimals)**

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

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Intro Product Quotient Chain Inversion Chain rule 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: , which has inside. , then Click here for a proof Next

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**If we put , we see that these two definitions are the same.**

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

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Go back to Chain Rule We have

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Go back to Chain Rule 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

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**Chain rule example Find . , so Intro Product Quotient Chain Inversion**

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**True or False? Intro Product Quotient Chain Inversion**

differentiates to Let: differentiates to Let: , and inside that, let: differentiates to Let: Next

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

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**Go back to Inversion Rule**

First note that , because we’re differentiating the function Then: by the chain rule, This is a function, so we can divide by it… We get:

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**Inversion Rule Example**

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

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Intro Product Quotient Chain Inversion Note, it is NOT TRUE that Next

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**Find at the specified values of y:**

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

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**More complicated example**

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

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**Chain rule on also gives . Then quotient rule gives:**

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

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Intro Product Quotient Chain Inversion Differentiate

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**Conclusion We can differentiate simple functions using the definition:**

Intro Product Quotient Chain Inversion 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

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