# Do arccoth(x) and arctanh(x) have the same derivatives?

• Benny
In summary, the derivatives of arccoth(x) and arctanh(x) are not the same. This is due to the difference in their domains, as well as the fact that the derivatives of ln(x) and ln(-x) are the same but their domains are disjoint.
Benny
Hello, I am wondering if arccoth(x) and arctanh(x) have the same derivatives? I ask this because I got:

$$\frac{d}{{dx}}arc\coth (x) = \frac{1}{{1 - x^2 }} = \frac{d}{{dx}}\arctan h(x)$$

I don't think their derivatives should be the same. Does it have anything to do with domain restrictions? Any help appreciated.

Well, you have:
$$artanh(x)=\frac{1}{2}ln(\frac{1+x}{1-x}), |x|<1, arcoth(x)=\frac{1}{2}ln(\frac{1+x}{x-1}), |x|>1$$

Note that, in general, the derivatives of ln(x) and ln(-x) with respect to "x" is the same; but their domains are disjoint.

Last edited:
Thanks for the help arildno.

## 1. What are arccoth(x) and arctanh(x)?

Arccoth(x) and arctanh(x) are inverse hyperbolic functions. Arccoth(x) is the inverse of the hyperbolic cotangent function, while arctanh(x) is the inverse of the hyperbolic tangent function.

## 2. Do arccoth(x) and arctanh(x) have the same derivatives?

Yes, arccoth(x) and arctanh(x) have the same derivatives. This means that the rate of change of both functions at any given point is equal. This can be shown mathematically through the use of the chain rule.

## 3. How do you find the derivative of arccoth(x) and arctanh(x)?

To find the derivative of arccoth(x), you can use the formula: d/dx arccoth(x) = 1 / (1 - x^2). For arctanh(x), the derivative can be found using the formula: d/dx arctanh(x) = 1 / (1 - x^2).

## 4. Are arccoth(x) and arctanh(x) continuous functions?

Yes, both arccoth(x) and arctanh(x) are continuous functions. This means that they have no breaks or gaps in their graphs and can be drawn without lifting the pencil from the paper.

## 5. How are arccoth(x) and arctanh(x) related to each other?

Arccoth(x) and arctanh(x) are related to each other as inverse functions. This means that they "undo" each other's actions. For example, if you take the hyperbolic cotangent of a number and then take the arccoth of that result, you will get the original number back.

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