Differentiate inverse trigonometric forms with guided chain rules. Enter coefficients, evaluate points, and compare results. Export reports quickly and visualize behavior from one page.
| Function | Input form | Derivative result | Sample at x = 0 |
|---|---|---|---|
| arcsin | arcsin(2x) | 2 / √(1 - 4x²) | 2 |
| arccos | arccos(3x - 1) | -3 / √(1 - (3x - 1)²) | Undefined |
| arctan | arctan(4x + 5) | 4 / (1 + (4x + 5)²) | 4 / 26 |
| arcsec | arcsec(2x + 3) | 2 / (|2x + 3|√((2x + 3)² - 1)) | 2 / (3√8) |
This calculator differentiates inverse trigonometric functions with a linear inner expression u(x) = ax + b. The chain rule says that the outer derivative must be multiplied by u'(x).
For arcsin(u), use u' / √(1 - u²). For arccos(u), use -u' / √(1 - u²). For arctan(u), use u' / (1 + u²). For arccot(u), use -u' / (1 + u²).
For arcsec(u), use u' / (|u|√(u² - 1)). For arccsc(u), use -u' / (|u|√(u² - 1)). The calculator also checks the real-domain conditions before reporting values.
Inverse trigonometric derivatives appear in calculus, differential equations, optimization, geometry, and engineering models. Students use them to solve related rates, tangent line problems, and integrals. Teachers use them for demonstrations. Analysts use them when transformations involve angle recovery and chain rule structure.
This calculator helps you move from formula memory to method. You can test different inner expressions, inspect where derivatives fail, compare values across a range, and export your work. The graph highlights how domain boundaries affect shape, slope, and valid evaluation points.
Because the tool keeps the input in the form u(x) = ax + b, it stays focused and dependable. That makes it useful for homework checks, worksheet preparation, revision sessions, and quick classroom examples. The result section gives a direct symbolic derivative and a numeric derivative at the chosen point.
It differentiates arcsin, arccos, arctan, arccot, arcsec, and arccsc when the inside expression is linear, written as ax + b.
Some inverse trigonometric derivatives only exist on restricted real domains. Boundary points can make square-root denominators zero, and outside points can make the original function nonreal.
Yes. It first differentiates the inside expression u(x) = ax + b, then multiplies that result by the standard inverse trigonometric derivative rule.
The inverse trigonometric function values are reported in radians. That is the standard setting for calculus formulas and derivatives.
Choose a range that includes your point of interest but avoids very large invalid intervals. A focused range usually makes the derivative behavior easier to read.
The CSV report includes the selected function, derivative formula, evaluated point, numeric outputs, status note, domain note, and the sample calculation table.
The PDF button captures the visible result report, including summary values, steps, and the calculation table, then saves it as a PDF file.
Yes. It is useful for checking symbolic structure, verifying numeric derivatives at a point, and reviewing domain limits before submitting your work.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.