Question

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

Answer 1

`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`

Answer 2

`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`