# How do you differentiate #f(x)=cos(3x)*(2/3tanx)# using the product rule?

According to the product rule:

Find each derivative separately. The first will require the chain rule:

Plug these back in:

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To differentiate the function ( f(x) = \cos(3x) \cdot \frac{2}{3}\tan(x) ) using the product rule, follow these steps:

- Identify the two functions being multiplied: ( u(x) = \cos(3x) ) and ( v(x) = \frac{2}{3}\tan(x) ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Differentiate each function separately: ( u'(x) = -3\sin(3x) ) and ( v'(x) = \frac{2}{3}\sec^2(x) ).
- Substitute the derivatives into the product rule formula.
- Simplify the expression to obtain 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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