How do you differentiate #f(x)=(xtanx)/(x^2x+3)# using the quotient rule?
There are 2 Answers first with the product rule the second with the quotient rule
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To differentiate ( f(x) = \frac{x \tan(x)}{x^2  x + 3} ) using the quotient rule, follow these steps:

Identify the numerator and denominator of the function.
 Numerator: ( x \tan(x) )
 Denominator: ( x^2  x + 3 )

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

Compute the derivatives of the numerator and denominator:
 ( u = x \tan(x) )
 ( u' = \tan(x) + x \sec^2(x) )
 ( v = x^2  x + 3 )
 ( v' = 2x  1 )
 ( u = x \tan(x) )

Substitute into the quotient rule formula: [ \frac{d}{dx} \left( \frac{x \tan(x)}{x^2  x + 3} \right) = \frac{(x \sec^2(x) + \tan(x))(x^2  x + 3)  (x \tan(x))(2x  1)}{(x^2  x + 3)^2} ]

Simplify the expression as needed.
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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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