What is the derivative of sqrt(x - 1)/sqrtx?

1 Answer
Jan 27, 2016

dy/dx = sqrt(x)/(2x^2sqrt(x-1))

Explanation:

To find the derivative of sqrt(x-1)/sqrt(x) we can try the following
sqrt(x-1)/sqrt(x) = sqrt((x-1)/x)
on simplifying further

=sqrt(1-1/x)

Let us use chain rule.

Let y=sqrt(u) and u=1-1/x

By chain rule

dy/dx = dy/(du) xx (du)/dx

y=sqrt(u)
dy/(du) = 1/(2sqrt(u))
Substituting back for u we get
dy/(du) = 1/(2sqrt(1-1/x))

Now we shall find our (du)/dx

u=1-1/x
(du)/dx = 0- (-1/x^2)
(du)/dx = 1/x^2

dy/dx = dy/(du) xx (du)/dx
dy/dx = 1/(2sqrt(1-1/x)) xx 1/x^2
dy/dx = 1/(2sqrt((x-1)/x)) xx 1/x^2

dy/dx = sqrt(x)/(2x^2sqrt(x-1))