Question

How do you differentiate `f(x)=(2cosx)/(x+1)`?

Answer 1

`(-2(x+1)sinx-2cosx)/(x+1)^2`

Explanation:

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

If `f(x)=(g(x))/(h(x))` then

`color(red)(bar(ul(|color(white)(a/a)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(a/a)|)))`
`color(blue)"----------------------------------------------------------------"`

here `g(x)=2cosxrArrg'(x)=-2sinx`

and `h(x)=x+1rArrh'(x)=1`
`color(blue)"-----------------------------------------------------------------"`
substitute these values into f'(x)

`rArrf'(x)=((x+1)(-2sinx)-2cosx(1))/(x+1)^2`

`=(-2(x+1)sinx-2cosx)/(x+1)^2`