Question

How do you find `(f^-1)'(a)` if `f(x)= x/sqrt{x-3}` and a = 4?

Answer 1

The function is not bijective, So, the bi-valued inverse is not differentiable. See the explanation..

Explanation:

y = f(x) is real for x >=3.

`y'=1/(x-3)-x/(2(x-3)^(3/2))=(x-6)/(2(x-3)^(3/2))`.

Inversely, bi-valued `x=(y (y+-sqrt(y^2-12)))/2`, with `|y|>=2sqrt3`

However, assigning y = 4, x = 4 or 12.for the two inverse forms.

Both are admissible....

The inverse is not unique for the given y = a = 4. The inverse is not

differentiable.