Question

Find the derivative of the function. \(y=\tanh ^{-1} \frac{x}{2}\)

Short answer

The derivative of the function \(y=\tanh^{-1} \frac{x}{2}\) is \(y' = \frac{2}{4 - x^2}\).

Step-by-step solution

Identify the function on which to apply the derivative

The given function is \(y=\tanh^{-1} \frac{x}{2}\). The derivative will be applied on it.

Apply the derivative

The derivative of \(y = \tanh^{-1} u\) is \(\frac{1}{1-u^2}\). Apply this to the given function such that \(u = \frac{x}{2}\). This gives, \(y' = \frac{1}{1-\left(\frac{x}{2}\right)^2}\). Do not forget to apply the chain rule. The derivative of \(\frac{x}{2}\) is \(\frac{1}{2}\). So, the derivative of \(y\) becomes, \(y' = \frac{1}{2(1-\left(\frac{x}{2}\right)^2)}\)

Simplify the expression

We have, \(y' = \frac{1}{2(1-\left(\frac{x}{2}\right)^2)} = \frac{1}{2\left(1-\frac{x^2}{4}\right)} = \frac{1}{2 - \frac{x^2}{2}} = \frac{2}{4 - x^2}\).