Problem 23 - Calculus Early Transcendentals

Question

Find \(d^{2} y / d x^{2}\) $$ y=\sin x \cos x $$

Step-by-Step Solution

Answer

\(\frac{d^2 y}{dx^2} = -2 \sin 2x \)

Step 1: Identify the Product Function

The function you're given is a product of two functions, \( y = \sin x \cdot \cos x \). You'll need to use the product rule to find the first derivative.

Step 2: Apply the Product Rule

The product rule states that for two functions \( u \) and \( v \), the derivative \( \frac{d}{dx}(uv) = u'v + uv' \).Set \( u = \sin x \) and \( v = \cos x \). Thus, \( u' = \cos x \) and \( v' = -\sin x \).

Step 3: Differentiate to Find \( \frac{dy}{dx} \)

Using the product rule:\[ \frac{dy}{dx} = \cos x \cdot \cos x + \sin x \cdot (-\sin x) = \cos^2 x - \sin^2 x \].

Step 4: Simplify \( \frac{dy}{dx} \) Using Trigonometric Identity

Use the trigonometric identity: \( \cos 2x = \cos^2 x - \sin^2 x \). Therefore, \( \frac{dy}{dx} = \cos 2x \).

Step 5: Differentiate \( \frac{dy}{dx} \) to Find \( \frac{d^2y}{dx^2} \)

Now differentiate \( \frac{dy}{dx} = \cos 2x \) with respect to \( x \). The derivative of \( \cos 2x \) is \( -2\sin 2x \), using the chain rule (where the derivative of \( 2x \) is \( 2 \)).

Key Concepts

Understanding the Product Rule

When dealing with derivatives of functions that are products of two or more functions, the product rule is an essential tool. The product rule states that if you have two functions, say \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

• \( \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

This means you differentiate each function separately, multiply by the other function, then sum the results. In the context of the original problem, we have \( u(x) = \sin x \) and \( v(x) = \cos x \). The derivatives are \( u'(x) = \cos x \) and \( v'(x) = -\sin x \). Applying the product rule combines these derivatives to find the first derivative of the product. This is a crucial step in finding the second derivative.

Using Trigonometric Identities

Simplifying expressions during derivative calculations often involves trigonometric identities. One common identity is the double angle identity:

• \( \cos(2x) = \cos^2(x) - \sin^2(x) \)

In the given problem, after applying the product rule, we end up with \( \cos^2 x - \sin^2 x \). This can be directly simplified using the double angle identity for cosine, resulting in \( \cos 2x \). This clever substitution not only simplifies the expression but also aids in further differentiation. Recognizing and applying these identities helps to streamline the process and make calculus problems more manageable.

The Chain Rule in Action

The chain rule is used in calculus to differentiate composite functions. When you have a function encapsulated within another function, like \( f(g(x)) \), you need the chain rule. It states:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

In the instance of this problem, we are tasked with differentiating \( \cos 2x \) to find the second derivative. Here, the outer function is \( \cos(x) \) and the inner function is \( 2x \). According to the chain rule, we differentiate \( \cos(x) \), getting \( -\sin(x) \), and multiply by the derivative of \( 2x \), which is 2. Therefore, the result of differentiating \( \cos 2x \) is \(-2\sin 2x \). Understanding the chain rule is vital when handling functions of this nature, where one function is nested within another.