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section41_material - Section 4.1 The Product and Quotient...

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MATH 110 BUSINESS CALCULUS CHAPTER 4 TECHNIQUES OF DIFFERENTIATION Section 4.1 The Product and Quotient Rules Product Rule If () f x and g x are differentiable functions of x, then so is their product ()() f xgx , and [] () () d f xg x dx ′′ =+ Example: Calculate dy dx . Simplify your answer. ( ) 32 1 yx x =− Solution: 22 3 244 24 13 1 2 3 3 2 3 5 dy d xx x x dx dx xxx ⎡⎤ = + ⎣⎦ =−−=
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Exercise: Calculate dy dx . You need not expand your answer. ( )( ) 22 4 yx x x x =+− Answer: 2 2 (4 )( ) (8 1)( ) )(1 2 ) d xx x x x x x x dx ⎡⎤ +−= + + + ⎣⎦ Exercise: Calculate dy dx . () 2 1 1 x x ⎛⎞ =+ + ⎜⎟ ⎝⎠ Answer: 2 23 1 1 11 1 2 1 dy d dx dx x + ⎢⎥ + + ⎜⎟⎜⎟
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Quotient Rule If () f x and g x are differentiable functions of x, then so is their quotient () () f xg x (provided () 0 gx ), and [] 2 ()() df x f x g x f x g x dx g x ′′ ⎡⎤ = ⎢⎥ ⎣⎦ Example: Calculate dy dx . Simplify your answer. 39 24 x y x = + Solution: 2 22 3 ( 2 4 ) 2 ( ) ( ) 61 2 8 3 0 (2 4) dy d x x x dx dx x x xx −+ == ++ +− +
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Applications Example:
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section41_material - Section 4.1 The Product and Quotient...

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