How do I find the derivative of #f(x)=2sin(x)cos(x)#?

In your case you have:

To find the derivative of ( f(x) = 2\sin(x)\cos(x) ), you can use the product rule of differentiation, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Applying this rule to ( f(x) = 2\sin(x)\cos(x) ), you differentiate ( \sin(x) ) and ( \cos(x) ) separately, then apply the product rule:

[ \frac{d}{dx} (2\sin(x)\cos(x)) = 2(\cos(x) \cdot \frac{d}{dx}(\sin(x)) + \sin(x) \cdot \frac{d}{dx}(\cos(x))) ]

Using the derivatives of ( \sin(x) ) and ( \cos(x) ), which are ( \cos(x) ) and ( -\sin(x) ) respectively, the expression simplifies to:

[ \frac{d}{dx} (2\sin(x)\cos(x)) = 2(\cos(x) \cdot \cos(x) - \sin(x) \cdot \sin(x)) ]

[ = 2(\cos^2(x) - \sin^2(x)) ]