Problem 57 - Calculus Single Variable

Question

Let \(f(x)=\sec ^{2}(x)-\tan ^{2}(x) .\) Use the Product Rule to differentiate each summand of \(f\). Simplify to show that \(f^{\prime}(x)=0\) for each \(x\) in the domain of \(f .\) How can you obtain this result more easily?

Step-by-Step Solution

Answer

Differentiate using the product rule and trigonometric identities: \(f'(x) = 0\). Recognize that \(f(x) = 1\).

Step 1: Identify and Differentiate the First Summand

The first summand is \(\sec^2(x)\). The derivative of \(\sec^2(x)\) using the chain rule is: \[\frac{d}{dx}[\sec^2(x)] = 2\sec(x)\cdot \sec(x) \tan(x) = 2\sec^2(x) \tan(x)\]

Step 2: Identify and Differentiate the Second Summand

The second summand is \(\tan^2(x)\). The derivative using the chain rule is: \[\frac{d}{dx}[\tan^2(x)] = 2\tan(x)\cdot \sec^2(x) = 2\tan(x)\sec^2(x)\]

Step 3: Combine the Derivatives

Combine the derivatives of both summands to find \(f'(x)\):\[f'(x) = 2\sec^2(x)\tan(x) - 2\tan(x)\sec^2(x)\]Both terms are identical but of opposite signs, so they cancel each other out: \[f'(x) = 0\]

Step 4: Verification Using Trigonometric Identities

Notice that \(\sec^2(x) - \tan^2(x) = 1\), a known trigonometric identity derived from \(\sec^2(x) - \tan^2(x) = \cos^{-2}(x) - \frac{\sin^2(x)}{\cos^2(x)} = 1\). Since \(f(x) = 1\), its derivative is clearly \(f'(x) = 0\).

Key Concepts

Product Rule

The Product Rule is an essential formula used in differentiation when you need to find the derivative of a product of two functions. It's expressed mathematically as:

• If you have two functions, say \( u(x) \) and \( v(x) \), the derivative of their product \( u(x) \cdot v(x) \) is given by: \( (uv)' = u'v + uv' \).

In the context of the given exercise, although we don't directly use the Product Rule because the terms are not products, this foundational knowledge helps us comfortably differentiate more complex expressions where multiple rules might apply at once, like the Chain Rule in combination with product-type structures.

Chain Rule

The Chain Rule is an indispensable tool in calculus. It helps us find derivatives of composite functions, those created when one function is nested inside another. Written succinctly, the Chain Rule states:

• If you have a composite function \( g(f(x)) \), the derivative is \( g'(f(x)) \cdot f'(x) \).

In our exercise, differentiating \( \sec^2(x) \) and \( \tan^2(x) \) serves as a perfect example of applying the Chain Rule. For both functions, the outer function is being raised to a power, while the inner function is the trigonometric angle expression itself. By first finding the derivative of the outer function and then multiplying it by the derivative of the inner one, we can accurately determine the derivative of these terms.

Trigonometric Identities

Trigonometric identities often simplify complex expressions, making differentiation and integration much easier. In our example, we see the utility of one such identity:

• \( \sec^2(x) - \tan^2(x) = 1 \).

This identity emerges from the fundamental identity \( 1 + \tan^2(x) = \sec^2(x) \), which itself is derived using Pythagorean identities. Recognizing and applying such identities saves significant effort, as it allows us to reduce seemingly complex functions to simple constants. For instance, when we observe that \( f(x) = 1 \), it follows that its derivative is necessarily zero due to its constant nature.

Secant

The secant function, represented as \( \sec(x) \), is the reciprocal of the cosine function: \( \sec(x) = \frac{1}{\cos(x)} \). Differentiating \( \sec(x) \) involves some complexity since we need to apply the Chain Rule along with basic knowledge of how reciprocal functions behave.To derive \( \sec(x) \), we use the knowledge that:

• \( \frac{d}{dx}\sec(x) = \sec(x)\tan(x) \).

This is essential for understanding how to tackle more complex expressions like \( \sec^2(x) \). In our exercise, the differentiation of \( \sec^2(x) \) was executed by recognizing \( \sec^2(x) \) as \((\sec(x))^2\), applying the Chain Rule directly.

Tangent

The tangent function, denoted as \( \tan(x) \), is one of the basic trigonometric functions and is defined as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Known for its repetitive pattern and vertical asymptotes, it behaves uniquely, especially under differentiation.The derivative of \( \tan(x) \) is:

• \( \frac{d}{dx}\tan(x) = \sec^2(x) \).

This derivative can be easily derived if you consider the tangent function as a division of sine and cosine and apply the Quotient Rule. In our particular exercise, though, we extend this by using the Chain Rule to effectively differentiate the square of the tangent function. Recognizing this transformation is key to simplification, ultimately revealing the derivative of the whole function as zero.