Question

How do you find the derivative of `1/sqrt (x-1)`?

Answer 1

The derivative is `=-1/(2(x-1)^(3/2))`

Explanation:

We need

`(x^n)'=nx^(n-1)`

Our function is `f(x)=1/sqrt(x-1)=(x-1)^(-1/2)`

`AA x in (1, +oo)`, `f(x) in (0,+oo)`

`f'(x)=((x-1)^(-1/2))'=-1/2*(x-1)^(-3/2)=-1/(2(x-1)^(3/2))`

Answer 2

`-1/(2sqrt((x-1)^3))`

Explanation:

`"express "y=1/(sqrt(x-1))=(x-1)^(-1/2)`

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

`"given "y=f(g(x))" then"`

`dy/dx=f'(g(x)xxg'(x)larr" chain rule"`

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

`rArrdy/dx=-1/2(x-1)^(-3/2)xxd/dx(x-1)`

`color(white)(rArrdy/dx)=-1/(2sqrt((x-1)^3))`