Chain, Product & Quotient Rules, Schemes and Mind Maps of Mathematics

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Chain, Product & Quotient Rules

• The product rule
• The quotient rule
• The chain rule
• Questions

The chain rule

• The chain rule is used to di ff erentiate a function that has a function within it. y = f (u) y = ( 2 x + 4 ) 3 u = f (x ) y = u 3 and u = 2 x + 4 dy du = 3 u 2 du dx

dy dx = 3 u 2 × 2 = 2 × 3 ( 2 x + 4 ) 2 dy dx

dy du

du dx dy dx = 6 ( 2 x + 4 ) 2

The quotient rule

• The quotient rule is used to di ff erentiate a function that is the division of two functions. f (x ) = u(x ) v (x ) f '(x ) = u 'v − v 'u v 2 dy dx

du dx × v

⎟+^

dv dx × u

u = 3 x − 5 f (x ) = 3 x − 5 4 x + 7 v = 4 x + 7 u ' = (^3) v ' = 4 f '(x ) = 3 ( 4 x + 7 ) − 4 ( 3 x − 5 ) ( 4 x + 7 ) 2

12 x + 21 − 12 x + 20 ( 4 x + 7 ) 2 f '(x ) =

( 4 x + 7 ) 2

Differentiation questions

• Find the derivative: (^) f (x ) = (x 2 + x )( 3 x − 1 ) u = x 2

• x , u ' = 2 x + 1 f '(x ) = ( 2 x + 1 )( 3 x − 1 ) + 3 (x 2
• x ) f '(x ) = 9 x 2
• 4 x + 1 f '(x ) = u 'v + v 'u v = 3 x + 1 , v ' = 3 f '(x ) = ( 6 x 2
• x + 1 ) + ( 3 x 2
• 3 x )

Differentiation questions (3)

• Find the derivative: y = 2 x (x − 6 ) 2 u = 2 x , v = (x − 6 ) 2 f '(x ) = u 'v − v 'u v 2 = 2 (x − 6 ) 2 − 2 (x − 6 )( 2 x ) ((x − 6 ) 2 ) 2 f '(x ) = u 'v − v 'u v 2 u ' = 2 , v ' = 2 (x − 6 ) f '(x ) = 2 (x − 6 ) 2 − 4 x(x − 6 ) (x − 6 ) 4 f '(x ) = − 2 (x + 6 ) (x − 6 ) 3

2 (x − 6 ) − 4 x (x − 6 ) 3

2 x − 12 − 4 x (x − 6 ) 3