Question

Multiple Choice Which of the following is \(\frac{d}{d x} \sec ^{-1}\left(x^{2}\right) ?\) (A) \(\frac{2}{x \sqrt{x^{4}-1}} \quad\) (B) \(\frac{2}{x \sqrt{x^{2}-1}} \quad\) (C) \(\frac{2}{x \sqrt{1-x^{4}}}\) (D) \(\frac{2}{x \sqrt{1-x^{2}}} \quad\) (E) \(\frac{2 x}{\sqrt{1-x^{4}}}\)

Short answer

The correct answer is (A) \(\frac{2}{x \sqrt{x^{4}-1}}\)

Step-by-step solution

Deriving the inverse secant function

First, it's necessary to know that the derivative of \(\sec^{-1}(x)\) is \(\frac{1}{|x| \sqrt{x^2 - 1}}\) for |x| > 1.

Applying the chain rule

The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is \(\sec^{-1}(x)\) and the inner function is \(x^{2}\). The chain rule then gives us \(\frac{d}{d x} \sec^{-1}\left(x^{2}\right) = \frac{1}{|x^{2}| \sqrt{(x^2)^{2} - 1}} \cdot 2x\).

Simplifying the expression

Simplifying that expression, we obtain \(\frac{2x}{|x^{2}| \sqrt{x^{4} - 1}}\). Since \(x^{2}\) is always positive, we can drop the absolute value, hence, the final result is \(\frac{2x}{x^{2} \sqrt{x^{4} - 1}}\).

Comparing with the options

None of the options exactly match our result. However, we can observe that the answer is structured in a way that suggests x is nonzero (since there is a x in the denominator). By canceling one instance of x both from the denominator and the numerator, we arrive at \(\frac{2}{x \sqrt{x^{4} - 1}}\), which is option A.

Key concepts

Chain Rule Calculus

The chain rule is a fundamental principle in calculus used for finding derivatives of composite functions. When a function is composed of an outer function and an inner function nested within, the chain rule allows us to differentiate this function systematically.

The formula for the chain rule states that if we have a composite function denoted as \(f(g(x))\), the derivative of this function with respect to \(x\) is \(f'(g(x)) \cdot g'(x)\). In other words, we multiply the derivative of the outer function, evaluated at the inner function, by the derivative of the inner function itself.

For example, if we have a function like \(h(x) = \sin(x^2)\), using the chain rule, we would first take the derivative of the outer function \(\sin(u)\), which is \(\cos(u)\), then replace \(u\) with \(x^2\). After that, we would multiply by the derivative of the inner function, which in this case is \(2x\). So, the derivative \(h'(x)\) would be \(2x \cdot \cos(x^2)\).

The chain rule is particularly useful in cases where the variable being differentiated is not directly in the function but raised to a power or under a root, as seen in the inverse secant derivative problem.

Derivative of Composite Functions

Understanding the derivative of composite functions is crucial for solving complex differentiation problems. A composite function is created when one function is applied to the result of another function. The notation \(f(g(x))\) exemplifies this, where \(f\) is the outer function and \(g\) is the inner function.

The main tool to tackle the differentiation of such functions is the chain rule. To elaborate, let's consider \(f(x) = \sec^{-1}(x^2)\). Here, \(\sec^{-1}(x)\) is the outer function and \(x^2\) is the inner function. While differentiating, we need to first differentiate the outer function as if the inner function were a simple variable, and then multiply it by the derivative of the inner function.

This approach not only simplifies the differentiation process but also helps in conceptualizing the relationship between nested functions, which is often a challenge when faced with more abstract mathematical expressions.

Simplifying Mathematical Expressions

Simplifying mathematical expressions is an essential skill in calculus to effectively communicate and comprehend solutions. The process generally involves reducing expressions to their simplest form by performing operations like canceling common factors, expanding and factoring polynomials, and using algebraic identities.

Consider the derivative \(\frac{2x}{|x^2| \sqrt{x^4 - 1}}\) from the inverse secant derivative problem. We can simplify this by recognizing that \(x^2\), being the square of any real number, is always positive, which allows us to remove the absolute value bars safely. Further, observing that \(x\) appears in both the numerator and the denominator, we can cancel out one instance of \(x\) to further simplify.

These steps reduce complex expressions and make it easier to compare with given options in a multiple-choice question. Simplification helps students to identify the correct answer quickly and is a strategy that aids in managing time efficiently during examinations and other timed assessments.