How do you find the first and second derivatives of #f(x)=(x)/(x^2+1)# using the quotient rule?
Below
The following is the quotient rule:
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Please see the explanation below
The rule of the quotient is
Here,
Consequently, the initial derivative is
Regarding the secondary derivative,
Consequently, the second derivative is
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To find the first derivative of ( f(x) = \frac{x}{x^2 + 1} ) using the quotient rule:

Apply the quotient rule: ( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v  uv'}{v^2} ), where ( u = x ) and ( v = x^2 + 1 ).

Find ( u' ) and ( v' ):
 ( u' = 1 )
 ( v' = 2x )

Plug ( u' ), ( v' ), ( u ), and ( v ) into the quotient rule formula: ( f'(x) = \frac{(1)(x^2 + 1)  (x)(2x)}{(x^2 + 1)^2} )

Simplify: ( f'(x) = \frac{x^2 + 1  2x^2}{(x^2 + 1)^2} ) ( f'(x) = \frac{1  x^2}{(x^2 + 1)^2} )
To find the second derivative, differentiate ( f'(x) = \frac{1  x^2}{(x^2 + 1)^2} ) using the quotient rule again:

Apply the quotient rule: ( f''(x) = \frac{d}{dx} \left( \frac{1  x^2}{(x^2 + 1)^2} \right) )

Follow the steps of the quotient rule with ( u = 1  x^2 ) and ( v = (x^2 + 1)^2 ).

Find ( u' ) and ( v' ):
 ( u' = 2x )
 ( v' = 2(x^2 + 1)(2x) )

Plug ( u' ), ( v' ), ( u ), and ( v ) into the quotient rule formula: ( f''(x) = \frac{(2x)((x^2 + 1)^2)  (1  x^2)(2(x^2 + 1)(2x))}{(x^2 + 1)^4} )

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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