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# lecture20hout - MATH 006 6 8 Quotient Rule Example Find the...

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MATH 006 Calculus and Linear Algebra (Lecture 20) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 8

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Outline 1 Product Rule 2 Quotient Rule Albert Ku (HKUST) MATH 006 2 / 8
Product Rule Product Rule Theorem (Product Rule) If y = f ( x ) = u ( x ) v ( x ) , then f ( x ) = u ( x ) v ( x ) + u ( x ) v ( x ) . Remark In general, ( u ( x ) v ( x )) = u ( x ) v ( x ). The product rule can be generalized as follows: ( u ( x ) v ( x ) w ( x )) = u ( x ) v ( x ) w ( x ) + u ( x ) v ( x ) w ( x ) + u ( x ) v ( x ) w ( x ) Albert Ku (HKUST) MATH 006 3 / 8

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Product Rule Example Find the derivative for each of the following functions: (a) f ( x ) = 2 x 2 e x (b) f ( x ) = ( x + 1)(ln x - 1) Solutions (a) f ( x ) = (2 x 2 ) e x + 2 x 2 ( e x ) = 4 xe x + 2 x 2 e x . (b) f ( x ) = ( x + 1) (ln x - 1) + ( x + 1)(ln x - 1) . Hence we have f ( x ) = 1 2 x - 1 2 (ln x - 1) + ( x + 1) 1 x . Albert Ku (HKUST) MATH 006 4 / 8
Product Rule Example Let f ( x ) = (2 x - 9)( x 2 + 6) (a) Find f ( x ) (b) Find the equation of the tangent line at x = 3 (c) Find the values of x where the tangent line is horizontal. Albert Ku (HKUST) MATH 006 5 / 8

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Quotient Rule Quotient Rule Theorem (Quotient Rule) If y = f ( x ) = u ( x ) v ( x ) , then dy dx = f ( x ) = v ( x ) u ( x ) - u ( x ) v ( x ) ( v ( x )) 2 . Remark In general, u ( x ) v ( x ) = u ( x ) v ( x ) . u ( x ) and v ( x ) in the above formula cannot be interchanged.

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Unformatted text preview: MATH 006 6 / 8 Quotient Rule Example Find the derivative for each of the following functions: (a) f ( x ) = x √ x +1 (b) f ( x ) = e x 2 x +1 Solutions (a) f ( x ) = x ( √ x + 1)-x ( √ x + 1) ( √ x + 1) 2 . Hence we have f ( x ) = √ x + 1-1 2 √ x ( √ x + 1) 2 = √ x + 2 2( √ x + 1) 2 . (b) f ( x ) = ( e x ) (2 x + 1)-e x (2 x + 1) (2 x + 1) 2 . Hence we have f ( x ) = e x (2 x + 1)-2 e x (2 x + 1) 2 = e x (2 x-1) (2 x + 1) 2 . Albert Ku (HKUST) MATH 006 7 / 8 Quotient Rule Example Find the derivative of each of the following functions: (a) f ( x ) = x 2 x 2-1 (b) g ( t ) = t 2-t t 3 + 1 (c) y = x 3 x 2 + 3 x + 4 (d) h ( x ) = 3 √ x x 2-3 (e) k ( s ) = 2 s-1 ( s 3 + 2)( s 2-3) Albert Ku (HKUST) MATH 006 8 / 8...
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lecture20hout - MATH 006 6 8 Quotient Rule Example Find the...

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