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

Factor the numerator first.

Next, remove a factor starting at the bottom.

As of right now, overall:

I double checked this answer, and it can be left this way. There is no more useful algebra to do.

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To differentiate the function ( f(x) = \frac{{1 - \sin^2(x)}}{{(1 - \sin(x))^2}} ) using the quotient rule:

- Identify ( u(x) ) as the numerator and ( v(x) ) as the denominator.
- Find the derivatives of ( u(x) ) and ( v(x) ).
- Apply the quotient rule: [ \frac{{d}}{{dx}}\left(\frac{{u(x)}}{{v(x)}}\right) = \frac{{u'(x)v(x) - u(x)v'(x)}}{{(v(x))^2}} ]
- Substitute the derivatives into the quotient rule formula.
- Simplify the expression.

The derivatives needed are:

- ( u'(x) = \frac{{d}}{{dx}}(1 - \sin^2(x)) )
- ( v'(x) = \frac{{d}}{{dx}}((1 - \sin(x))^2) )

After finding ( u'(x) ) and ( v'(x) ), substitute them into the quotient rule formula and simplify to get the final result.

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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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