Second Derivatives of Inverse Trigonometric Compositions

Why These Second Derivatives Need Care

A composition such as \(y=\arcsin(x^2)\) combines two layers of change: the inner algebraic function changes with \(x\), and the inverse trigonometric function responds to that change. Differentiating once uses the chain rule. Differentiating a second time requires differentiating both the outer derivative and the inner derivative.

That second differentiation creates two terms. Keeping both is essential, and checking the domain is just as important: the usual derivative formulas for inverse sine and inverse cosine require their inputs to lie strictly between \(-1\) and \(1\).

Inverse Trigonometric Functions and Their Domains

Here, \(\arcsin u\), \(\arccos u\), and \(\arctan u\) mean the principal inverse functions. Their output ranges are, respectively, \([-\frac{\pi}{2},\frac{\pi}{2}]\), \([0,\pi]\), and \((-\frac{\pi}{2},\frac{\pi}{2})\). The first two functions accept inputs in \([-1,1]\); the third accepts every real input.

For inputs strictly inside the domain, their derivative formulas are:

Suppose \(g\) has a second derivative on an open interval. To apply the second-derivative formulas below throughout that interval, require \(-1<g(x)<1\) for \(\arcsin(g(x))\) and \(\arccos(g(x))\). For \(\arctan(g(x))\), there is no corresponding restriction on the value of \(g(x)\). These conditions ensure the relevant first-derivative formulas can be differentiated there.

The Second-Derivative Formulas

Let \(y\) be one of these compositions. Applying the chain rule once gives its first derivative. Applying the chain rule and product rule to that result gives the second derivative:

The formulas are similar, but the inverse tangent denominator has \(1+g(x)^2\), while inverse sine and cosine involve \(1-g(x)^2\). The cosine formula is the negative of the sine formula for the same inner function.

Why Two Terms Appear

For inverse sine, first differentiate once using \(u=g(x)\):

To differentiate again, apply the product rule. The derivative of the first factor is \(g''(x)\). For the second factor, use the chain rule:

Therefore the complete second derivative is

This derivation explains the two contributions: one comes from differentiating \(g'(x)\), and one comes from differentiating the factor that depends on \(g(x)\). For inverse cosine, the first derivative is the negative of the inverse-sine first derivative, so differentiating it gives the negative of both terms above.

For inverse tangent, the same product-rule structure applies:

Multiplying the derivative of the second factor by \(g'(x)\), and adding the derivative of the first factor, gives the stated inverse-tangent formula.

Worked Examples

Domain Checks and a Common Mistake

For \(\arcsin(g(x))\) and \(\arccos(g(x))\), first check that \(g(x)\) lies in \([-1,1]\) wherever the composition is being considered. To use the displayed second-derivative formulas at an input, the stronger condition \(-1<g(x)<1\) is required. At \(g(x)=1\) or \(g(x)=-1\), the denominator in the derivative formulas is zero, so these formulas do not apply there. A particular composition might have special behavior at an endpoint, but it must be analyzed separately rather than evaluated by substituting into a formula with a zero denominator.

For \(\arctan(g(x))\), there is no restriction on \(g(x)\)'s real value, although \(g\) still needs to be twice differentiable at the input. In every case, evaluate the inner function and its first two derivatives before substituting into the corresponding formula.

For example, with \(y=\arcsin(x^2)\), an incomplete calculation might keep only

This omits the derivative of \((1-g(x)^2)^{-1/2}\). The correct calculation includes both contributions:

At \(x=\frac12\), the incomplete expression gives \(\frac{8}{\sqrt{15}}\), while the correct value is \(\frac{136}{15\sqrt{15}}\). The missing term is \(\frac{16}{15\sqrt{15}}\), not zero.

A Reliable Calculation Process

These second derivatives arise whenever a model or function includes an inverse trigonometric response to a changing algebraic input. The reliable approach is not to memorize a single expanded expression, but to track the inner function, apply the appropriate chain rule, and preserve both terms at the second differentiation.