Given that y=(4x+1)^3sin 2x , find dy/dx .

So this function is the product of two functions of x, so we use the product rule to differentiate it. The rule states if y=uv, dy/dx=(du/dx)v+(dv/dx)u. In this function we assign u=(4x+1)³ and v=sin2x. When we differentiate u we need to use the chain rule, as there is a function within a function, which gives us (3(4x+1)²)x4 which is equal to 12(4x+1)². When we differentiate v we get 2cos2x, again using chain rule. So we plug these values into the formula which gives us dy/dx=12(4x+1)²Sin2x + 2(4x+1)³Cos2x