Solve inverse hyperbolic derivatives with clear steps and checks. Export results, review domains, and learn formulas through practical examples today.
| Function | a | b | c | x | u = bx + c | Derivative result |
|---|---|---|---|---|---|---|
| arsinh(u) | 1 | 2 | 1 | 0.5 | 2 | 2 / √5 ≈ 0.894427 |
| arcosh(u) | 1 | 3 | 2 | 1 | 5 | 3 / (√4 × √6) ≈ 0.612372 |
| artanh(u) | 2 | 1 | 0 | 0.2 | 0.2 | 2 / (1 - 0.04) ≈ 2.083333 |
The calculator applies the chain rule to inverse hyperbolic functions. First, it builds the inner expression u = bx + c. Next, it finds u′ = b. Then it multiplies the known inverse hyperbolic derivative by u′ and the outer multiplier a.
| Function | Derivative |
|---|---|
| arsinh(u) | u′ / √(u² + 1) |
| arcosh(u) | u′ / (√(u - 1) × √(u + 1)) |
| artanh(u) | u′ / (1 - u²) |
| arcoth(u) | u′ / (1 - u²) |
| arsech(u) | -u′ / (u × √(1 - u²)) |
| arcsch(u) | -u′ / (|u| × √(1 + u²)) |
Inverse hyperbolic derivatives appear in calculus, engineering, and mathematical modeling. Students often remember the direct hyperbolic derivatives but forget the inverse forms. This calculator makes the process easier. It shows the exact derivative rule, the chain rule structure, and the evaluated numeric answer.
The page handles six common inverse hyperbolic functions. These are arsinh, arcosh, artanh, arcoth, arsech, and arcsch. You can also apply a multiplier and a linear inner expression. That means the tool is not limited to a plain x input. It works for expressions such as 3·arsinh(2x + 1).
Inverse hyperbolic functions have important real-number restrictions. Arcosh needs an inner value above one. Artanh needs an inner value between negative one and one. Arsech needs an inner value between zero and one. This calculator checks those conditions before returning a real-valued derivative result.
The output explains each major part. It computes the inner term, the derivative of the inner term, and the outer derivative formula. Then it multiplies everything together. This is useful for homework review, exam practice, and quick verification while solving manual problems.
The example table helps you compare different inverse hyperbolic forms. The formula section gives a compact reference sheet. The export options make it easy to save your result for class notes, tutoring sessions, or worksheet preparation. The overall layout stays clean and simple for fast use.
It finds the derivative of a selected inverse hyperbolic function with a linear inner term. It also evaluates the derivative at your chosen x value.
The calculator supports arsinh, arcosh, artanh, arcoth, arsech, and arcsch. These are common inverse hyperbolic forms used in advanced calculus work.
Yes. It uses u = bx + c, computes u′ = b, and multiplies by the correct inverse hyperbolic derivative rule automatically.
Some inverse hyperbolic functions require restricted real domains. The calculator blocks invalid real inputs so the result stays mathematically meaningful.
Yes. The inputs accept integers and decimals. You can also choose the precision level for the displayed derivative values.
The value a is the outer multiplier in y = a × f(bx + c). It scales the final derivative directly.
The CSV button exports the visible calculation fields and result values. This is useful for records, assignments, and quick sharing.
The PDF button opens the browser print dialog for saving the result section as a PDF. It is simple and works widely.