Given that y=(4x+1)^3*sin(2x) , find dy/dx
y=(4x+1)^3sin(2x) - this is a product of two functions of x. It can be rewritten as y = u(x)v(x) ; where u(x) = (4x+1)^3 and v(x) = sin(2x) Using the product rule: dy/dx = u'(x)v(x) + v'(x)u(x) where the ' (prime) notation denotes the differential with respect to x u'(x) = 34(4x+1)^2 and v'(x) = 2cos(2x) using either substitution or simplification rules for both Therefore, using product rule, dy/dx=[ 34*(4x+1)^2 ] * [ sin(2x) ] + [ 2cos(2x) ] * [ (4x+1)^3 ] which simplifies to: dy/dx = 2(4x+1)^3cos(2x) + 12(4x+1)^2
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