How do you differentiate #f(x)=xcosx# using the product rule?
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To differentiate ( f(x) = x \cos(x) ) using the product rule:
- Identify the functions being multiplied: ( u(x) = x ) and ( v(x) = \cos(x) ).
- Apply the product rule formula: ( (uv)' = u'v + uv' ).
- Find the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = 1 ) (since the derivative of ( x ) with respect to ( x ) is 1) ( v'(x) = -\sin(x) ) (using the derivative of ( \cos(x) ) which is ( -\sin(x) )).
- Plug the derivatives into the product rule formula: ( f'(x) = (1)(\cos(x)) + (x)(-\sin(x)) ).
- Simplify: ( f'(x) = \cos(x) - x\sin(x) ).
So, the derivative of ( f(x) = x \cos(x) ) with respect to ( x ) is ( \cos(x) - x\sin(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.
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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