How do you differentiate #f(x)=(xsinx)/(x^2tanx)# using the quotient rule?
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To differentiate ( f(x) = \frac{x  \sin(x)}{x^2  \tan(x)} ) using the quotient rule, follow these steps:

Identify the numerator ( u(x) = x  \sin(x) ) and the denominator ( v(x) = x^2  \tan(x) ).

Apply the quotient rule: [ f'(x) = \frac{u'(x)v(x)  u(x)v'(x)}{(v(x))^2} ]

Differentiate the numerator and denominator: [ u'(x) = 1  \cos(x) ] [ v'(x) = 2x  \sec^2(x) ]

Substitute the values into the quotient rule formula: [ f'(x) = \frac{(1  \cos(x))(x^2  \tan(x))  (x  \sin(x))(2x  \sec^2(x))}{(x^2  \tan(x))^2} ]

Simplify the expression.
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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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