How do you use the quotient rule to find the derivative of #y=x/(x^2+1)# ?
Quotient Rule
Make all of the necessary substitutions
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To use the quotient rule to find the derivative of ( y = \frac{x}{x^2 + 1} ), follow these steps:

Identify ( u ) and ( v ) in the function ( y = \frac{u}{v} ). In this case, ( u = x ) and ( v = x^2 + 1 ).

Apply the quotient rule formula: [ y' = \frac{u'v  uv'}{v^2} ]

Calculate ( u' ) and ( v' ): [ u' = 1 ] [ v' = 2x ]

Plug the values into the formula: [ y' = \frac{(1)(x^2 + 1)  (x)(2x)}{(x^2 + 1)^2} ]

Simplify the expression: [ y' = \frac{x^2 + 1  2x^2}{(x^2 + 1)^2} ] [ y' = \frac{1  x^2}{(x^2 + 1)^2} ]
Therefore, the derivative of ( y = \frac{x}{x^2 + 1} ) is ( y' = \frac{1  x^2}{(x^2 + 1)^2} ).
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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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