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

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To differentiate ( f(x) = (cosx + x)(3x^3 - e^x) ) using the product rule:

- Identify the two functions being multiplied together: ( u(x) = cosx + x ) and ( v(x) = 3x^3 - e^x ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Calculate the derivatives of ( u(x) ) and ( v(x) ):
- ( u'(x) = -sinx + 1 ) (derivative of ( cosx + x )).
- ( v'(x) = 9x^2 - e^x ) (derivative of ( 3x^3 - e^x )).

- Substitute the derivatives and original functions into the product rule formula:
- ( f'(x) = (-sinx + 1)(3x^3 - e^x) + (cosx + x)(9x^2 - e^x) ).

- Simplify the expression if needed.

So, the derivative of ( f(x) ) using the product rule is ( f'(x) = (-sinx + 1)(3x^3 - e^x) + (cosx + x)(9x^2 - e^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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