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y = 2sin(x)cos(x)

Use the product rule: uv' + vu' where u is 2sin(x) and v is cos(x) to find first derivative:

y' = 2sin(x)(-sin(x)) + cos(x)2cos(x)

Simplify:

y' = 2cos2(x)-2sin2(x)

y' = 2(cos2(x)-sin2(x))

Use trig identity cos(2x) = cos2(x)-sin2(x):

y' = 2cos(2x)

Take second derivative using chain rule:

y'' = 2(-sin(2x)cos(2x))

Simplify:

y'' = -2sin(2x)(2)

Simplify:

y'' = -4sin(2x)y'' = -4sin(2x)

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Q: What is the second derivative of y equals 2sinx cosx?
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