Chain Rule To differentiate composite functions we have to use the Chain rule. Composite functions are functions of a function.
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1 Differentiation Workshop: Differentiation the rules Topics Coered: Chain rule Proct rule Quotient rule Combinations Chain Rule To differentiate composite functions we hae to use the Chain rule. Composite functions are functions of a function. If we hae y f(t) and t g(), then the deriatie of y with respect to is, y is a function in terms of t t is a function in terms of Eample : Differentiate y ( ) with respect to Let t y t t t 6t 6( ) Eample : Differentiate y with respect to y ( ) Let t y t ½ ½t -½ Dr Mundeep Gill Brunel Uniersity
2 5 ½t -½ t 5 5 ( ) ( ) Eample : Differentiate y sin ( + 9) with respect to. Let t Using the Chain rule: y sin t cos t cos (t) (6 + 9) (6 + 9)cos ( + 9) Questions (Chain rule): Differentiate the following functions with respect to.. y ( + ) 5. y cos (5 + 6). y + e. y ln ( + ) (Solutions on page 8)
3 The Proct Rule Consider y u, where u and are functions of, then Eample : Differentiate y 7e Let u 7 e 7 e Using the proct rule: 7e + e 7e ( + ) Eample : Differentiate y sin () Let u sin() cos() Using the proct rule: sin() + cos() (sin() + cos()) Eample : Differentiate y e ln(5) Let u e e ln(5)
4 Using the proct rule: e ln(5) + e ( ) e ln 5 + Questions (Proce rule): Differentiate the following functions with respect to :. y 5ln(6). y e 5 ( + ). y 5 cos(). y e ln(7) (Solutions on page 9) The Quotient Rule Consider y u, where u and are functions of, then u Eample : Differentiate y + u + u Using the quotient rule: ( + ) ( ) ( + )
5 [( + ) ( ) ] ( + ) ( + + ) ( + ) ( ) ( + ) ( ) + Eample : Differentiate y ln + 5 u ln () u Using the quotient rule: ( + 5) ( + ) ( + 5) ( + 5 ) ( + ) ( + 5) 5 ln 5 ln Eample : Differentiate y 5e e + u 5e e + 5e e + u Using the quotient rule: 5
6 5e ( e + ) ( 5e )( e + ) ( e + ) 5e + 0e ( 5e + 0e e ) ( e + ) 5e + 0e 5e ( e + ) 7e + 0e 7e ( e + ) + Questions (Quotient rule): Differentiate the following functions with respect to :. y. y. y. y cos ln ( ) e sin + cos (Solutions on page 0) Combinations Sometimes you may need to use a combination of the rules to differentiate a function. Eample : Differentiate y 5 ( + ) t + t t Solution: Let u 5 ( + ) 5 ( + ) t ( + ) 6
7 Using the proct rule: 5 ( + ) + 5 ( + ) ( + ) [5( + ) + ] ( + ) [ ] ( + ) (9 + 5) ln( + ) Eample : Differentiate y t + u ln t t Solution: u ln ( + ) - + u Using the quotient rule: ( -) ln( + ) ( ) + ( ) ( -) ( + ln ) ( + ) ( + )( ) t t + Questions (Combinations): Differentiate the following functions with respect to :. y ( + )( + ). y ( ) + (Solutions on page ) 7
8 Solutions (Chain rule):. y ( + ) 5 Let t + y t t (6 + ) 5t 5t (6 + ) 5( + ) (6 + ). y cos (5 + 6) Let t y cos t 5 -sin t -sin t (5 ) -5 sin(t) -5 sin (5 + 6). y + e Let t + y e t + ( + ) e + e t. y ln ( + ) Let t + y ln t + t 8
9 ( + ) t ( + ) ( + ) Solutions (Proct rule):. y 5ln(6) Let u 5 5 ln(6) Using the proct rule: 5ln() + 5 5ln() + 5 5(ln() + ). y e 5 ( + ) Let u e 5 + 5e 5 Using the proct rule: 5e 5 ( + ) + e 5 e 5 (5( + ) + ) e 5 ( ) e 5 (0 + 9). y 5 cos() Let u 5 5 cos() -sin() 9
10 Using the proct rule: 5 cos() 5 sin() [5cos() sin()]. y e ln(7) Let u e e ln(7) Using the proct rule: e ln(7) + e ( ) e ln 7 + Solutions (Quotient rule):. y u - u Using the quotient rule: ( ) ( ) ( - ) - + ( - ) - ( ) - ( ) ( - ) - 0
11 . y cos u cos () -sin u Using the quotient rule: cos + ( cos ) sin ( cos + sin) cos. y ln ( ) u ln () - u Using the quotient rule: ( -) ln( ) ( ) ( - ) ln( ) ( ) - ln ( ) ( )
12 . y e sin + cos u e sin () + cos () e cos () sin () u Using the quotient rule: ( sin + cos )( e ) ( e )( cos sin) e e e ( sin + cos ) [ ( sin + cos ) ( cos sin) ] ( sin + cos ) ( sin + cos cos + sin) ( sin + cos ) ( sin + cos ) ( sin + cos ) Solutions (Combinations):. y ( + )( + ) Let u ( + ) ( + ) ( + ). 9( + ) Using the proct rule: ( + ) + 9( + )( + ) ( + ) [( + ) + 9( + )] ( + ) [ ] ( + ) [ ]
13 . y ( ) + u Using the quotient rule: u ( + ) ( + ) u ( + ) ( + ) ( + ) [ ] ( + ) ( + ) ( + ) ( + ) ( + ) + ( + ) + ( + ) ( + ) ( + )
![) ( + ) [ ] ( + ) ( + ) ( + ) (](/docs-images/46/21517479/images/page_13.jpg "+ ) ( + ) + ( + ) + ( + ) ( + )")
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( ) = ( ) = {,,, } β ( ), < 1 ( ) + ( ) = ( ) + ( )
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