Differentiation – Product, Quotient and Chain Rules Department of Mathematics University of Leicester

Content Introduction Product Rule Quotient Rule Chain Rule Inversion Rule

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

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

Go back to Product Rule Let . Then:

Go back to Product Rule

Product rule example Find . Intro Product Quotient Chain Inversion Next

Differentiate these: Intro Product Quotient Chain Inversion Take: Next

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

Go back to Quotient Rule Let . Then:

Go back to Quotient Rule

Go back to Quotient Rule

Quotient rule example Find . Intro Product Quotient Chain Inversion Next

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

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

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.

Go back to Chain Rule We have .

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 .

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

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

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

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: .

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

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

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

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

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

Intro Product Quotient Chain Inversion Differentiate

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