How do you differentiate #f(x)=(xxsinx+1)/(x^23x)# using the quotient rule?
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To differentiate ( f(x) = \frac{x  x \cdot \sin(x) + 1}{x^2  3x} ) using the quotient rule:

Identify ( u(x) ) and ( v(x) ). ( u(x) = x  x \cdot \sin(x) + 1 ) ( v(x) = x^2  3x )

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

Find the derivatives of ( u(x) ) and ( v(x) ): ( u'(x) = 1  (\sin(x) + x \cdot \cos(x)) ) ( v'(x) = 2x  3 )

Substitute the derivatives and original functions into the quotient rule formula: ( f'(x) = \frac{(x^2  3x)(1  (\sin(x) + x \cdot \cos(x)))  (x  x \cdot \sin(x) + 1)(2x  3)}{(x^2  3x)^2} )

Simplify the expression if necessary.
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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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