Calculator
Plotly Graph
The graph plots the derivative values over the selected x-range. Invalid domain points are skipped automatically.
Example Data Table
| Example Function | Derivative Formula | Derivative Value | Domain Note |
|---|---|---|---|
| asinh(1) | d/dx[asinh(x)] = 1 / sqrt(x^2 + 1) |
0.707107 | All real x |
| acosh(2) | d/dx[acosh(x)] = 1 / (sqrt(x - 1) * sqrt(x + 1)) |
0.577350 | Derivative is real for x > 1 |
| atanh(0.5) | d/dx[atanh(x)] = 1 / (1 - x^2) |
1.333333 | Derivative is real for -1 < x < 1 |
| acoth(2) | d/dx[acoth(x)] = 1 / (1 - x^2) |
-0.333333 | Derivative is real for |x| > 1 |
| asech(0.5) | d/dx[asech(x)] = -1 / (x * sqrt(1 - x^2)) |
-2.309401 | Derivative is real for 0 < x < 1 |
| acsch(2) | d/dx[acsch(x)] = -1 / (|x| * sqrt(1 + x^2)) |
-0.223607 | Derivative is real for x ≠ 0 |
Formula Used
This calculator applies standard derivative rules for inverse hyperbolic functions. Each rule is evaluated at a chosen x-value after checking the valid derivative domain. That keeps the output mathematically correct and prevents invalid square roots or division by zero.
d/dx[asinh(x)] = 1 / sqrt(x^2 + 1)d/dx[acosh(x)] = 1 / (sqrt(x - 1) * sqrt(x + 1))d/dx[atanh(x)] = 1 / (1 - x^2)d/dx[acoth(x)] = 1 / (1 - x^2)d/dx[asech(x)] = -1 / (x * sqrt(1 - x^2))d/dx[acsch(x)] = -1 / (|x| * sqrt(1 + x^2))
For functions such as acosh(x), atanh(x), acoth(x), asech(x), and acsch(x), the derivative only exists on certain real-number intervals. The calculator checks those intervals before showing a numeric answer.
How to Use This Calculator
- Select the inverse hyperbolic function you want to differentiate.
- Enter the x-value where you want the derivative evaluated.
- Choose how many decimal places to display.
- Set a graph half range and number of sample points.
- Press the calculate button to show the result above the form.
- Review the derivative formula, evaluated expression, domain note, and working steps.
- Use the CSV or PDF buttons to export the result table.
About This Calculator
Inverse hyperbolic functions appear in calculus, differential equations, complex analysis, and applied mathematics. Their derivatives often look similar to inverse trigonometric derivatives, but the sign patterns and domain restrictions differ. That makes checking the correct rule important, especially during exams, assignments, or symbolic work.
This calculator helps you move from the general derivative rule to a numeric value at a chosen point. It also highlights domain restrictions, which matter because some inverse hyperbolic derivatives are only real on certain intervals. For example, atanh(x) only accepts values strictly between negative one and positive one, while acoth(x) works outside that interval. A quick domain check prevents invalid evaluation.
The graph gives a visual view of how the derivative behaves around your selected x-value. That can help you see growth, sign changes, and steep regions more clearly. The example table gives ready reference values, while the export tools make it easy to save results for notes, worksheets, or classroom review.
FAQs
1) What does this calculator return?
It returns the derivative rule for a selected inverse hyperbolic function and evaluates that derivative at your chosen x-value. It also shows domain notes, a graph, working steps, and export buttons.
2) Why are some x-values rejected?
Some inverse hyperbolic derivatives are only real on restricted domains. When your x-value falls outside that interval, the calculator marks the result as outside the derivative domain instead of forcing an invalid output.
3) Why do atanh(x) and acoth(x) have the same algebraic derivative?
They share the expression 1 / (1 - x^2), but their valid real domains differ. Atanh(x) uses -1 < x < 1, while acoth(x) uses |x| > 1. Domain changes matter.
4) Why does acsch(x) use absolute value?
The derivative of acsch(x) includes |x| so the formula stays correct for both positive and negative inputs. Without absolute value, the sign behavior would be wrong across the real line.
5) What does the graph show?
The graph plots derivative values across the selected interval centered near your chosen x-value. Points outside the valid derivative domain are skipped automatically so the plot stays meaningful.
6) Can I export the result?
Yes. After calculation, use the CSV button to download a table file or the PDF button to save a printable version of the result summary.
7) Is this useful for homework checking?
Yes. It is helpful for checking formulas, verifying numeric answers, and reviewing domain restrictions. It is best used as a checking tool alongside your own written derivation steps.
8) Does it differentiate custom full expressions?
No. This version evaluates built-in inverse hyperbolic derivative rules at a point. It is designed for targeted checking, study, graphing, and export rather than general symbolic expression parsing.