# How do you differentiate #f(x)= (1 - sin^2x)/(x-cosx) # using the quotient rule?

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To differentiate the function ( f(x) = \frac{1 - \sin^2(x)}{x - \cos(x)} ) using the quotient rule, you follow these steps:

- Identify ( u(x) ) as the numerator ( 1 - \sin^2(x) ) and ( v(x) ) as the denominator ( x - \cos(x) ).
- Compute ( u'(x) ) and ( v'(x) ).
- Apply the quotient rule: ( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ).
- Substitute the values of ( u'(x) ), ( u(x) ), ( v'(x) ), and ( v(x) ) into the formula.
- Simplify the expression to get the derivative of the function.

The derivative of ( f(x) ) using the quotient rule is:

[ f'(x) = \frac{(2\sin(x)\cos(x) - 1)(x - \cos(x)) - (1 - \sin^2(x))(1 + \sin(x))}{(x - \cos(x))^2} ]

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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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