Second Derivative of Inverse Trigonometric Function Calculator

Calculator

What This Calculator Does

Purpose and use cases. It computes the second derivative of a composition of an inverse trigonometric function with an inner function u(x). Typical tasks include curvature analysis, Taylor expansions, locating inflection points, or verifying manual work.

Symbolic vs numeric modes. Two numerical routes are implemented: (i) direct central differencing of f(x) = g(u(x)) where g is the selected inverse trig, and (ii) a chain-rule formulation f'' = G'(u)[u']^2 + G(u)u'' where G = dg/du. The interface also shows the symbolic base formulas so you can write down closed-form results when u(x) is known symbolically.

Input Requirements

Function formats supported. Choose one of: arcsin, arccos, arctan, arcsec, arccsc, arccot. The inner function u(x) may use: sin, cos, tan, sec, csc, cot, asin, acos, atan, asec, acsc, acot, sqrt, abs, exp, ln, log, log10, constants pi, e, plus arithmetic and parentheses.

Radians assumption and variable domain. All trig functions use radians. Domain constraints are enforced numerically at x₀ (e.g., |u| ≤ 1 for arcsin/arccos; |u| ≥ 1 for arcsec/arccsc). Near boundaries, choose a small h to avoid stepping outside the domain.

Quick Start

Select an inverse trig function.
Enter u(x) using the supported syntax (e.g., x^2 + 1 or sin(x)).
Set x and the step size h, then compute.
Compare the two numeric routes and consult the symbolic formulas below.

Differentiation Rules Recap

First derivatives of inverse trig functions (with respect to u):

\[ \frac{{d}}{{du}}\arcsin(u) = \frac{{1}}{\sqrt{{1-u^2}}},\quad \frac{{d}}{{du}}\arccos(u) = -\frac{{1}}{\sqrt{{1-u^2}}},\\ \frac{{d}}{{du}}\arctan(u) = \frac{{1}}{{1+u^2}},\quad \frac{{d}}{{du}}\arccot(u) = -\frac{{1}}{{1+u^2}},\\ \frac{{d}}{{du}}\arcsec(u) = \frac{{1}}{{|u|\sqrt{{u^2-1}}}},\quad \frac{{d}}{{du}}\arccsc(u) = -\frac{{1}}{{|u|\sqrt{{u^2-1}}}}. \]

Chain rule and product/quotient rules used. For f(x)=g(u(x)), one has \[ f'(x)=G(u)u'(x)\quad\text{{and}}\quad f''(x)=G'(u)[u'(x)]^2 + G(u)u''(x), \] where \(G(u)=g'(u)\) as listed above.

Second Derivative Formulas (Base Cases)

Let \(u=u(x)\), \(u'=\tfrac{{du}}{{dx}}\), \(u''=\tfrac{{d^2u}}{{dx^2}}\). Then:

\[ \frac{{d^2}}{{dx^2}}\arcsin(u) = \frac{{u''(1-u^2) + u(u')^2}}{{(1-u^2)^{3/2}}},\quad \frac{{d^2}}{{dx^2}}\arccos(u) = -\frac{{u''(1-u^2) + u(u')^2}}{{(1-u^2)^{3/2}}}. \]

\[ \frac{{d^2}}{{dx^2}}\arctan(u) = \frac{{u''(1+u^2) - 2u(u')^2}}{{(1+u^2)^2}},\quad \frac{{d^2}}{{dx^2}}\arccot(u) = -\frac{{u''(1+u^2) - 2u(u')^2}}{{(1+u^2)^2}}. \]

For \(\arcsec\) and \(\arccsc\) (principal branches, \(|u|>1\)) one convenient form via \(f''=G'(u)[u']^2+G(u)u''\) is: \[ G(u)=\frac{{\pm 1}}{{|u|\sqrt{{u^2-1}}}},\quad G'(u) = \mp\frac{{\operatorname{{sgn}}(u)}}{{|u|^2\sqrt{{u^2-1}}}} \mp \frac{{u}}{{|u|(u^2-1)^{3/2}}}, \] with the upper sign for \(\arcsec\) and the lower sign for \(\arccsc\).

When using numeric evaluation, the calculator handles the absolute value and sign terms automatically.

FAQs

Does it use degrees or radians? Radians.
Why do the two numeric routes differ slightly? Finite differencing and rounding errors. Try a smaller h and stay within domain.
Can I enter compositions like arcsin(sin(x))? Yes, but remember domain restrictions and principal values.
How do I enter logarithms? Use ln() for natural log or log10() for base 10.
What if u(x) is undefined at x0? You’ll see an evaluation error or NaN; adjust x0 or the expression.
Are arcsec and arccsc supported? Yes; domains require |u| ≥ 1 and sign handling is built-in.
Can I export results? Copy outputs directly; the page is self-contained and print-friendly.
What accuracy should I expect? Central differences are second-order in h. Reasonable h (e.g., 1e-5) works well for smooth functions.

Related Calculators