# How do you differentiate #f(x)= 2x*sinx*cosx#?

Use the product rule:

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To differentiate ( f(x) = 2x \cdot \sin(x) \cdot \cos(x) ), you can use the product rule. The product rule states that if you have a function ( u(x) ) multiplied by another function ( v(x) ), the derivative of the product is ( u'(x) \cdot v(x) + u(x) \cdot v'(x) ). Applying this rule to ( f(x) ):

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

Find ( u'(x) ) and ( v'(x) ), then substitute them into the product rule formula:

( u'(x) = 2 ) ( v'(x) = (\sin(x) \cdot \cos(x))' = \cos^2(x) - \sin^2(x) )

Now, apply the product rule:

( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) )

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

Simplify the expression:

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

This is the derivative of ( f(x) ).

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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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