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