`y = tanh^-1(x/2)` Find the derivative of the function

Derivative of a function f with respect to x is denoted as `f'(x)` or `y'` .

To solve for derivative of y (y') for the given problem: `y = tanh^(-1)(x/2)` , 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 = x/2` then   `du=1/2dx` or `(du)/(dx)=1/2`

Then, the problem becomes:

`d/(dx)(tanh^(-1)(x/2))=(1/2)/((1-(x/2)^2))`

                             ` =(1/2)/((1-x^2/4))`

                             ` = (1/2)/(((4-x^2)/4))`

                             ` = (1/2)*4/((4-x^2))`

                               `= 2/(4-x^2)`

Then, the final answer is:

`d/(dx)(tanh^(-1)(x/2))=2/(4-x^2)`  where `|x/2|lt1`