# How do you differentiate #cosx/(sinx-2)#?

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To differentiate the given function ( \frac{\cos(x)}{\sin(x) - 2} ), you can use the quotient rule, which states that if you have a function ( \frac{f(x)}{g(x)} ), its derivative is given by:

[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2} ]

Where ( f'(x) ) is the derivative of ( f(x) ) with respect to ( x ), and ( g'(x) ) is the derivative of ( g(x) ) with respect to ( x ).

Now, applying this rule to the given function ( \frac{\cos(x)}{\sin(x) - 2} ), we have:

[ f(x) = \cos(x) ] [ g(x) = \sin(x) - 2 ]

[ f'(x) = -\sin(x) ] [ g'(x) = \cos(x) ]

Now, substituting these into the quotient rule formula, we get:

[ \frac{d}{dx} \left( \frac{\cos(x)}{\sin(x) - 2} \right) = \frac{(-\sin(x) \cdot (\sin(x) - 2) - \cos(x) \cdot \cos(x))}{(\sin(x) - 2)^2} ]

Simplify the expression to get the final answer.

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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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