How do you differentiate #f(x) = (cosx)/(sinx)# using the quotient rule?
Apply the rule of quotient.
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To differentiate ( f(x) = \frac{\cos(x)}{\sin(x)} ) using the quotient rule:

Apply the quotient rule, which states: [ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x)  u(x)v'(x)}{(v(x))^2} ]

Identify ( u(x) = \cos(x) ) and ( v(x) = \sin(x) ).

Compute the derivatives:
 ( u'(x) = \sin(x) )
 ( v'(x) = \cos(x) )

Apply the quotient rule formula: [ f'(x) = \frac{(\sin(x))(\sin(x))  (\cos(x))(\cos(x))}{(\sin(x))^2} ]

Simplify the expression: [ f'(x) = \frac{\sin^2(x)  \cos^2(x)}{\sin^2(x)} ]

Use the trigonometric identity ( \sin^2(x) + \cos^2(x) = 1 ): [ f'(x) = \frac{1}{\sin^2(x)} ]

Rewrite the expression in terms of (\csc(x)): [ f'(x) = \csc^2(x) ]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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