How do you find the first and second derivatives of #f(x)=(x)/(x^2+1)# using the quotient rule?

Answer 1

Below

#f(x)=x/(x^2+1)#
#f'(x)= ((x^2+1)(1)-x(2x))/(x^2+1)^2#
#f'(x)=(x^2+1-2x^2)/(x^2+1)^2#
#f'(x)=(1-x^2)/(x^2+1)^2#
#f''(x)=((x^2+1)^2(-2x)-(1-x^2)*(x^2+1)(2x))/(x^2+1)^4#
#f''(x)=(-2x(x^2+1)-2x(1-x^2))/(x^2+1)^3#
#f''(x)=(-2x^3-2x-2x+2x^3)/(x^2+1)^3#
#f''(x)=(-4x)/(x^2+1)^3#

The following is the quotient rule:

#f(x)=u/v#
#f'(x)=(vu'-uv')/v^2#
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Answer 2

Please see the explanation below

The rule of the quotient is

#(u/v)'=(u'v-uv')/(v^2)#

Here,

#u=x#, #=>#, #u'=1#
#v=x^2+1#, #=>#, #v'=2x#

Consequently, the initial derivative is

#f'(x)=(1(x^2+1)-x(2x))/(x^2+1)^2#
#=(1-x^2)/(x^2+1)^2#

Regarding the secondary derivative,

#u=(1-x^2)#, #=>#, #u'=-2x#
#v=(x^2+1)^2#, #=>#, #v'=4x(x^2+1)#

Consequently, the second derivative is

#f''(x)=(-2x(x^2+1)^2-4x(1-x^2)(x^2+1))/(x^2+1)^4#
#=((x^2+1)(-2x(x^2+1)-4x(1-x^2)))/(x^2+1)^4#
#=(2x^3-6x)/(x^2+1)^3#
#=(2x(x^2-3))/(x^2+1)^3#
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Answer 3

To find the first derivative of ( f(x) = \frac{x}{x^2 + 1} ) using the quotient rule:

  1. 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 ).

  2. Find ( u' ) and ( v' ):

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

  4. 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:

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

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

  3. Find ( u' ) and ( v' ):

    • ( u' = -2x )
    • ( v' = 2(x^2 + 1)(2x) )
  4. 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} )

  5. Simplify the expression.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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