Find dy/dx for (4x + 4)(sin x).

Answer 1

d [ ( 4 x + 4 ) sin ( x ) ] / dx =

4 ∙ [ d ( x + 1 ) sin ( x ) ] / dx =

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Use the product rule:

d ( u v ) / dx = v du / dx + u dv / dx

where

u = x + 1

v = sin ( x )
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4 ∙ [ sin ( x ) ∙ d ( x + 1 ) / dx + ( x + 1 ) ∙ d sin ( x ) / dx ] =

4 ∙ [ sin ( x ) ∙ 1 + ( x + 1 ) ∙ cos ( x ) ] =

4 ∙ [ sin ( x ) + ( x + 1 ) ∙ cos ( x ) ]

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Remark

d ( x + 1 ) / dx = 1

becouse

d ( x + 1 ) / dx =

d ( x ) / dx + d ( 1 ) / dx =

1 + 0 = 1
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