# What is the derivative of #f(x)=cos^2x*cos2x#?

Here, we observe

Thus,

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Let's use the trigonometric identity:

Now we can take the derivative using the chain rule:

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To find the derivative of ( f(x) = \cos^2(x) \cdot \cos(2x) ), you can use the product rule.

The product rule states that if ( u(x) ) and ( v(x) ) are differentiable functions, then the derivative of their product ( u(x) \cdot v(x) ) with respect to ( x ) is ( u'(x) \cdot v(x) + u(x) \cdot v'(x) ).

Apply the product rule:

[ f'(x) = (\cos^2(x))' \cdot \cos(2x) + \cos^2(x) \cdot (\cos(2x))' ]

Find the derivatives of ( \cos^2(x) ) and ( \cos(2x) ):

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

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

Plug these derivatives back into the product rule formula:

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

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

So, the derivative of ( f(x) = \cos^2(x) \cdot \cos(2x) ) is ( f'(x) = -2\cos(x) \sin(x) \cos(2x) - 2\cos^2(x) \sin(2x) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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