# What is the derivative of #sinx(sinx+cosx)#?

The answer is

First, use the Product Rule to say

Next, expand this out to write

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To find the derivative of ( \sin(x)(\sin(x) + \cos(x)) ), we can use the product rule.

Let ( u = \sin(x) ) and ( v = \sin(x) + \cos(x) ). Then, apply the product rule:

[ \frac{d}{dx} [u \cdot v] = u'v + uv' ]

Where ( u' ) and ( v' ) are the derivatives of ( u ) and ( v ) respectively.

[ u' = \cos(x) ] [ v' = \cos(x) - \sin(x) ]

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

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

[ = 2\sin(x)\cos(x) + \sin(2x) ]

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