# What is the derivative of #sin^2(x) * cos^2(x)#?

By the chain and product rules,

In order to evaluate this derivative, we need to use both the product and chain rules.

Continuing with the product rule, we add the left- and right-hand derivatives we calculated above together, so our final answer is:

This can be simplified in several ways, but one simplified version of the derivative may be:

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To find the derivative of ( \sin^2(x) \cdot \cos^2(x) ), we can use the product rule of differentiation.

Let ( u = \sin^2(x) ) and ( v = \cos^2(x) ).

Then, ( \frac{du}{dx} = 2\sin(x)\cos(x) ) and ( \frac{dv}{dx} = -2\sin(x)\cos(x) ).

Applying the product rule, we have:

[ \frac{d}{dx}(\sin^2(x)\cdot \cos^2(x)) = \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx} ]

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

[ = 2\sin(x)\cos(x)\cos^2(x) - 2\sin^3(x)\cos(x) ]

[ = 2\sin(x)\cos(x)\cos^2(x) - 2\sin^3(x)\cos(x) ]

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

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

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