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The Chain Rule , finding the derivative of a composition. A composition of functions is a function , u(x) inside another function, f(u), making a function of x, f(u(x)). For example f (x) The inside function is x 3 x 20 3 u ( x ) x 3 x 20 . 3 The outside function is f ( u ) u If x changes , then u changes and then f changes. Consider two linear functions, u( x) 1 f ( u ) 3 u The composition is x 2 1 f ( u ( x )) 3 x 2 3 x 2 The slopes multiplied. The chain rule says the same is true for nonlinear functions. The Chain Rule: df df du dx Examples in which f ' (u )u ' ( x ) du dx u( x) x 3 : 2 a ) h ( x ) ( x 3) 2 5 h ' ( x ) 5( x 3) ( 2 x ) 2 4 Here h ( x ) f ( u ( x )) where f ( u ) u 5 f ' ( u ) u ' ( x ) 5u ( 2 x ) 5( x 3) ( 2 x ) 4 b) f ( x) 1 x 3 2 2 ( x 3) 2 2 1 2x 2 f ' ( x ) ( x 3) 2 x 4 x 2 3 2 f(x) in part b could also be done using the quotient rule. More examples : Find the derivative of each function. 1) f ( x ) ( x 1) 2) g ( x ) ( 2 x 5) 3) h ( x ) ( x 1) 4) f (x) 3 3 2 x ( x 1) 2 2 4 5) g(x) 2x 3 5x 7 2 6) A snowball is a perfect sphere with volume given by V (r) 4 r 3 cubic cm. 3 Find the rate of change of the volume with respect to the radius. If the radius is decreasing by 2 cm per min. due to melting, how fast is the volume decreasing when the radius is 30 cm? 7) The revenue as a function of x=number of units sold in the month is given by R ( x ) 0 . 2 x 40 x . If the quantity sold is decreasing by 12 units per month, 2 at what rate is revenue changing with respect to t=time in months if x=150?