Home > Calculus > Quotient Rule

How do you use the quotient rule to find the derivative of #y=x/(x^2+1)# ?

Answer 1

Ezekiel Alexander

Quotient Rule

#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2#

Make all of the necessary substitutions

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

#g(x) = x# #g'(x)= 1#

#h(x)=x^2+1# #h'(x)=2x+0# #h'(x)=2x#

#(h(x))^2=(x^2+1)^2#

#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#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Christopher Allers

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} ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.