# How do you differentiate #f(x)=(1/(3x))*x*e^x-x*cosx# using the product rule?

Please see below.

Simplify first

We do not need the product rule for the first term, although you may use it if you like.

For the second term, we need the product rule. (unless we do something unusual.)

Putting it all together,

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To differentiate the function ( f(x) = \frac{1}{3x} \cdot x \cdot e^x - x \cdot \cos(x) ) using the product rule, follow these steps:

- Identify the two functions being multiplied together: ( u(x) = \frac{1}{3x} \cdot x \cdot e^x ) and ( v(x) = -x \cdot \cos(x) ).
- Apply the product rule: ( (uv)' = u'v + uv' ).
- Compute the derivatives of ( u(x) ) and ( v(x) ):
- ( u'(x) = \left(\frac{1}{3x} \cdot x \cdot e^x\right)' )
- ( v'(x) = (-x \cdot \cos(x))' ).

- Calculate the derivatives using the product rule and the chain rule where necessary.
- Add the two results together to get the derivative of the original function ( 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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