What Is the Derivative of Arctanh(x/2)?

[ultima9999]

Yeah, I was working through this problem and it differs from the answer that my friend got.

Using implicit differentiation, find the derivative of $$\mbox{arc}\tanh \frac{x}{2}$$ and state the domain for which the derivative applies

$$\begin{align*} y = \arctanh \frac{x}{2} \\ \Leftrightarrow x = 2 \tanh y \end{align*}$$

$$\frac{d}{dx}x = \frac{d}{dx}2 \tanh y$$
$$\Rightarrow 1 = 2\ \mbox{sech}^2 y \cdot \frac{dy}{dx}$$
$$\Rightarrow \frac{dy}{dx} = \frac{1}{2\ \mbox{sech}^2 y}$$
$$\Rightarrow \frac{dy}{dx} = \frac{1}{2 - 2 \tanh^2 y}$$
$$\Rightarrow \frac{dy}{dx} = \frac{1}{2 - 2 \frac{x^2}{4}}$$
$$\Rightarrow \frac{dy}{dx} = \frac{1}{2 - \frac{x^2}{2}}$$

 

[0rthodontist]

The derivative of what now? d(x/2)/dx = 1/2. I assume you mean tanh^-1(x/2).

Yeah, you're right.