Evaluate the given function

Question

Let

f(x) = √In(t)dt

(a) Evaluate f'(10) =

(b) Evaluate (f ^-1)'(0) =

Answer 1

f(t ) = [lnt]^1/2 Domain of f the interval (1,∞ )

f^' (t) = 1 / [2t√lnt ]

1. f^' (10) = 1 / [20√ln10 ]
2. Here comes the interesting stuff . Since f (1 ) = 0 , making use of the Theorem of Derivative of the Inverse Function ( f^-1)^'(0) = 1 / f´(1) .But f´(1) is undefined . With this specific function the line tangent to the graph at x=1 is perpendicular to the x-axis. Hence the graph of the inverse at x=0 must have a tangent line horizontal to the x-axis, which of course has slope zero.. Then

( f^-1)^'(0) = 0..You can verify that by finding the inverse of f, being ( f^-1)^´(x) = e^{x^2} , then

( f^-1)^´(x) =2x e^{x^2} and ( f^-1)^´(0)=0