Question

If `f(x)= sqrt(x-2` and `g(x) = 1/x`, what is `f'(g(x))`?

Answer 1

`-1/(2x^2sqrt(1/x-2)`

Explanation:

Before differentiating we require to establish f(g(x))

Substitute x `=1/x"into f(x)"`

`rArrf(g(x))=f(1/x)=sqrt(1/x-2)`

now differentiate using the `color(blue)"chain rule combined with power rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(f(g(x)))=f'(g(x))g'(x))color(white)(a/a)|)))........ (A)`

Express `sqrt(1/x-2)" as " (x^-1-2)^(1/2)`
`"--------------------------------------------------"`
`f(g(x))=(x^-1-2)^(1/2)rArrf'(g(x))=1/2(x^-1-2)^(-1/2)`

and `g(x)=x^-1-2rArrg'(x)=-1x^(-2)=-x^(-2)`
`"--------------------------------------------------------"`
Substitute these values into (A)

`rArrf'(g(x))=1/2(x^-1-2)^(-1/2).(-x^-2)`

Expressing the answer with positive indices

`rArrf'(g(x))=-1/(2x^2sqrt(1/x-2)`