Calculus of a Single Variable, Chapter 5, 5.8, Section 5.8, Problem 67

Derivative of a function f with respect to x is denoted as f'(x) or y' .
To solve for derivative of y or (y') for the given problem: y = tanh^(-1)(sqrt(x)) , we follow the basic derivative formula for inverse hyperbolic function:
d/(dx)(tanh^(-1)(u))= ((du)/(dx))/(1-u^2) where |u|lt1 .
Let: u =sqrt(x)
Apply the Law of Exponent: sqrt(x) = x^(1/2)
Solve for the derivative of u using the Power Rule for derivative: d/(dx)x^n=n*x^(n+1) * d(x)
Then,
du=1/2x^(1/2-1)*1dx
du=1/2x^(-1/2) dx
Apply the Law of Exponent:
x^(-n)= 1/x^n.
du=1/(2x^(1/2)) dx
Rearrange into:
(du)/(dx)=1/(2x^(1/2))
(du)/(dx)=1/(2sqrt(x))
Apply the derivative formula, we get:
d/(dx)(tanh^(-1)(sqrt(x)))= ((1/(2sqrt(x))))/((1-(sqrt(x))^2))
=((1/(2sqrt(x))))/((1-x))
=(1/(2sqrt(x)))*1/((1-x))
=1/(2sqrt(x)(1-x))
Final answer:
d/(dx)(tanh^(-1)(sqrt(x)))=1/(2sqrt(x)(1-x))