Question

How do you differentiate `f(x) = (sqrtx)/(-x^2-2x+1)` using the quotient rule?

Answer 1

`f'(x)=(x^2+1)/(sqrtx(-x^2-2x+1)^2)`

Explanation:

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

Given `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)|)))........ (A)`
`color(blue)"------------------------------------------------------------"`

here `g(x)=sqrtx=x^(1/2)rArrg'(x)=1/2x^(-1/2)`

and `h(x)=-x^2-2x+1rArrh'(x)=-2x-2`
`color(blue)"------------------------------------------------------------"`
substitute these values into (A)

`f'(x)=((-x^2-2x+1).1/2x^(-1/2)-x^(1/2)(-2x-2))/(-x^2-2x+1)^2`

simplifying the numerator.

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

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

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