Calculator Form
This tool evaluates common inverse hyperbolic derivative rules and applies the chain rule using the provided inner derivative value.
Derivative Graph
This graph plots derivative values across sample input points using the selected inverse hyperbolic function and the chosen inner derivative value.
Formula Used
General chain rule idea: If the function is written as an inverse hyperbolic function of u(x), differentiate the outer inverse function and multiply by u'(x).
- d/dx asinh(u) = u' / √(u² + 1)
- d/dx acosh(u) = u' / [√(u - 1) √(u + 1)]
- d/dx atanh(u) = u' / (1 - u²)
- d/dx acoth(u) = u' / (1 - u²)
- d/dx asech(u) = -u' / [u √(1 - u²)]
- d/dx acsch(u) = -u' / [|u| √(1 + u²)]
How to Use This Calculator
- Select the inverse hyperbolic function you want to differentiate.
- Enter the evaluated value of the inner function u(x).
- Enter the corresponding value of the inner derivative u'(x).
- Optionally type an expression label for reference.
- Press Calculate Derivative to view the result above the form.
- Review the worked steps, graph, and example table.
- Use the export buttons to save the output as CSV or PDF.
Example Data Table
| Function | u(x) | u'(x) | Derivative Rule | Example Result |
|---|---|---|---|---|
| asinh(u) | 1.5 | 1 | u' / √(u² + 1) | 0.554700 |
| acosh(u) | 2.0 | 3 | u' / [√(u - 1) √(u + 1)] | 1.732051 |
| atanh(u) | 0.5 | 2 | u' / (1 - u²) | 2.666667 |
| acoth(u) | 2.0 | 1 | u' / (1 - u²) | -0.333333 |
| asech(u) | 0.4 | 1 | -u' / [u √(1 - u²)] | -2.727724 |
| acsch(u) | 2.0 | 1 | -u' / [|u| √(1 + u²)] | -0.223607 |
Frequently Asked Questions
1. What does this calculator compute?
It computes numeric derivative values for common inverse hyperbolic functions after applying the chain rule to the entered inner function value and derivative.
2. Does it perform symbolic differentiation?
It focuses on evaluated derivative results and guided formulas. It explains the rule used, but it does not parse full symbolic algebra expressions automatically.
3. Why can some inputs return an undefined result?
Each inverse hyperbolic function has a domain restriction. If your chosen value falls outside that valid domain, the derivative is undefined for real-number output.
4. What is u'(x) in this tool?
u'(x) is the derivative of the inner expression. The calculator multiplies the outer inverse hyperbolic derivative rule by this entered chain rule factor.
5. Can I use negative values?
Yes, for functions whose domains allow them. For example, asinh accepts all real values, while asech requires values between zero and one.
6. What does the graph show?
The graph shows sampled derivative values across a range of inputs for the selected rule. Invalid domain points are skipped automatically.
7. What is the benefit of the example table?
It provides quick reference cases for each supported inverse hyperbolic derivative, helping you compare formulas, domains, and expected numeric outputs.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet-friendly output and the PDF button for a printable snapshot of the current page content.