How do you differentiate #f(x)=(x-2)/(x^4+1)#?

2 Answers
Sep 6, 2017

#f'(x)=(8x^3-3x^4+1)/(x^4+1)^2#

Explanation:

#"differentiate using the "color(blue)"quotient rule"#

#"given "f(x)=(g(x))/(h(x))" then"#

#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larr" quotient rule"#

#g(x)=x-2rArrg'(x)=1#

#h(x)=x^4+1rArrh'(x)=4x^3#

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

#color(white)(rArrf'(x))=(8x^3-3x^4+1)/(x^4+1)^2#

Sep 6, 2017

#f'(x) = (8x^3-3x^4+1)/(x^4+1)^2 #

Explanation:

We could simply use the u/v rule of differentiation, which is

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

where,

#u# and #v# are diffrentiable functions and #u'# and #v'# are their derivatives.

In this question,

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

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

On differentiating the function we have,

#implies f'(x) = (( d/dx ( x-2))(x^4+1) - (x-2)(d/dx (x^4+1)))/(x^4+1)^2 #

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

#impliesf'(x) = (8x^3-3x^4+1)/(x^4+1)^2 #