lecture20hout

lecture20hout - MATH 006 6 / 8 Quotient Rule Example Find...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 006 Calculus and Linear Algebra (Lecture 20) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 8
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Outline 1 Product Rule 2 Quotient Rule Albert Ku (HKUST) MATH 006 2 / 8
Background image of page 2
Product Rule Product Rule Theorem (Product Rule) If y = f ( x ) = u ( x ) v ( x ) , then f 0 ( x ) = u ( x ) v 0 ( x ) + u 0 ( x ) v ( x ) . Remark In general, ( u ( x ) v ( x )) 0 6 = u 0 ( x ) v 0 ( x ). The product rule can be generalized as follows: ( u ( x ) v ( x ) w ( x )) 0 = u 0 ( x ) v ( x ) w ( x ) + u ( x ) v 0 ( x ) w ( x ) + u ( x ) v ( x ) w 0 ( x ) Albert Ku (HKUST) MATH 006 3 / 8
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 0 ( x ) = (2 x 2 ) 0 e x + 2 x 2 ( e x ) 0 = 4 xe x + 2 x 2 e x . (b) f 0 ( x ) = ( x + 1) 0 (ln x - 1) + ( x + 1)(ln x - 1) 0 . Hence we have f 0 ( x ) = 1 2 x - 1 2 (ln x - 1) + ( x + 1) 1 x . Albert Ku (HKUST) MATH 006 4 / 8
Background image of page 4
Product Rule Example Let f ( x ) = (2 x - 9)( x 2 + 6) (a) Find f 0 ( 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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Quotient Rule Quotient Rule Theorem (Quotient Rule) If y = f ( x ) = u ( x ) v ( x ) , then dy dx = f 0 ( x ) = v ( x ) u 0 ( x ) - u ( x ) v 0 ( x ) ( v ( x )) 2 . Remark In general, ± u ( x ) v ( x ) ² 0 6 = u 0 ( x ) v 0 ( x ) . u ( x ) and v ( x ) in the above formula cannot be interchanged. Albert Ku (HKUST)
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 12/28/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

Page1 / 8

lecture20hout - MATH 006 6 / 8 Quotient Rule Example Find...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online