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

Answer 1

#f'(x)=2sinxcosx+2xcos^2x-2xsin^2x#

Use the product rule:

#f=ghk# => #f'=g'hk+gh'k+ghk'#
With: #g=2x# => #g'=2x# #h=sinx# => #h'=cosx# #k=cosx# => #k'=-sinx#
We then have: #f'(x)=2sinxcosx+2xcos^2x-2xsin^2x#
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Answer 2

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

#f'(x)=(2x)' cdot (sin(x) cdot cos(x))+2x cdot (sin(x) cdot cos(x))'#
#(2x)'=2#
#(sin(x) cdot cos(x))'=sin(x)' cdot cos(x)+sin(x) cdot cos(x)'# #=cos(x) cdot cos(x)+sin(x) cdot (-sin(x))# #=cos^2(x)-sin^2(x)#
#f'(x)=2sin(x)cos(x)+2x(cos^2(x)-sin^2(x))#
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Answer 3

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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Answer from HIX Tutor

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.

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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