A proof of the Product Rule appears below. Provide a justification for each step. a. d d x [ f ( x ) ⋅ g ( x ) ] = lim h → 0 f ( x + h ) g ( x + h ) − f ( x ) g ( x ) h Definition of derivative b. = lim h → 0 f ( x + h ) g ( x + h ) − f ( x + h ) g ( x ) + f ( x + h ) g ( x ) − f ( x ) g ( x ) h Adding and subtracting the same quantify is the same as adding 0. c. = lim h → 0 f ( x + h ) g ( x + h ) − f ( x + h ) g ( x ) h + lim h → 0 f ( x + h ) g ( x ) − f ( x ) g ( x ) h The limit of a sum is the sum of the limits. d. = lim h → 0 [ f ( x + h ) ⋅ g ( x + h ) − g ( x ) h ] + lim x → 0 [ g ( x ) ⋅ f ( x + h ) − f ( x ) h ] e. = f ( x ) ⋅ lim h → 0 g ( x + h ) − g ( x ) h + g ( x ) ⋅ lim h → 0 f ( x + h ) − f ( x ) h The limit of a product is the product of the limit and lim h → 0 f ( x + h ) = f ( x ) . f. f ( x ) ⋅ g ' ( x ) + g ( x ) ⋅ f ' ( x ) Definition of derivative g. f ( x ) ⋅ [ d d x g ( x ) ] + g ( x ) ⋅ [ d d x f ( x ) ] Using Leibniz notation
A proof of the Product Rule appears below. Provide a justification for each step. a. d d x [ f ( x ) ⋅ g ( x ) ] = lim h → 0 f ( x + h ) g ( x + h ) − f ( x ) g ( x ) h Definition of derivative b. = lim h → 0 f ( x + h ) g ( x + h ) − f ( x + h ) g ( x ) + f ( x + h ) g ( x ) − f ( x ) g ( x ) h Adding and subtracting the same quantify is the same as adding 0. c. = lim h → 0 f ( x + h ) g ( x + h ) − f ( x + h ) g ( x ) h + lim h → 0 f ( x + h ) g ( x ) − f ( x ) g ( x ) h The limit of a sum is the sum of the limits. d. = lim h → 0 [ f ( x + h ) ⋅ g ( x + h ) − g ( x ) h ] + lim x → 0 [ g ( x ) ⋅ f ( x + h ) − f ( x ) h ] e. = f ( x ) ⋅ lim h → 0 g ( x + h ) − g ( x ) h + g ( x ) ⋅ lim h → 0 f ( x + h ) − f ( x ) h The limit of a product is the product of the limit and lim h → 0 f ( x + h ) = f ( x ) . f. f ( x ) ⋅ g ' ( x ) + g ( x ) ⋅ f ' ( x ) Definition of derivative g. f ( x ) ⋅ [ d d x g ( x ) ] + g ( x ) ⋅ [ d d x f ( x ) ] Using Leibniz notation
Solution Summary: The author explains how the value of different functions is used to define the derivatives of the function.
Precalculus Enhanced with Graphing Utilities (7th Edition)
Precalculus Enhanced with Graphing Utilities
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