Problem 192 - Calculus Volume 1

Question

For the following exercises, find \(\frac{d^{2} y}{d x^{2}}\) for the given functions. $$y=\sin x \cos x$$

Step-by-Step Solution

Answer

The second derivative is \( \frac{d^2 y}{dx^2} = -2\sin(2x) \).

Step 1: Identify the Product Rule

The given function is a product of two functions: \( \sin x \) and \( \cos x \). To find the first derivative, we need to apply the product rule. The product rule states that if \( y = u(x) \cdot v(x) \), then \( \frac{dy}{dx} = u'(x)v(x) + u(x)v'(x) \). Here, let \( u = \sin x \) and \( v = \cos x \).

Step 2: Differentiate the Components

First, find \( u'(x) \) and \( v'(x) \). For \( u(x) = \sin x \), we have \( u'(x) = \cos x \). For \( v(x) = \cos x \), we have \( v'(x) = -\sin x \).

Step 3: Apply the Product Rule

Substitute the derivatives into 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 Using Trigonometric Identity

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

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

Now we need to differentiate \( \cos(2x) \) to find the second derivative. Since the derivative of \( \cos u \) is \( -\sin u \cdot u' \), we have \( \frac{d}{dx}\left[ \cos(2x) \right] = -\sin(2x) \cdot (2) = -2\sin(2x) \).

Step 6: Conclusion

Thus, the second derivative of the given function is \( \frac{d^2 y}{dx^2} = -2\sin(2x) \).

Key Concepts

Understanding the Product Rule

The product rule is an essential concept of calculus that helps us differentiate functions that are products of two or more smaller functions. When dealing with a function like \(y = \sin x \cdot \cos x\), which is a product of \(\sin x\) and \(\cos x\), the product rule comes into play. The rule states that if you have a product of two functions, \(u(x)\) and \(v(x)\), the derivative of their product \(y = u(x) \cdot v(x)\) is given by:\[\frac{dy}{dx} = u'(x) \cdot v(x) + u(x) \cdot v'(x).\]This formula tells us to first differentiate one function while keeping the other constant, and then switch—differentiating the other while keeping the first function constant—and finally add the two results.

• Differentiate \(u(x)\), keep \(v(x)\) constant: \(u'(x) \cdot v(x)\)
• Keep \(u(x)\) constant, differentiate \(v(x)\): \(u(x) \cdot v'(x)\)

In our case, letting \(u(x) = \sin x\) and \(v(x) = \cos x\), we have:

• \(u'(x) = \cos x\)
• \(v'(x) = -\sin x\)

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

Trigonometric Identities and Simplification

Trigonometric identities are vital tools in calculus, especially for simplifying the results of differentiation. In the previous step, we found the first derivative as \(\cos^2 x - \sin^2 x\). Here, a useful trigonometric identity comes into play: \(\cos^2 x - \sin^2 x = \cos(2x)\).This identity simplifies our derivative significantly, reducing it to \(\cos(2x)\). Simplifying expressions in calculus often involves recognizing standard identities like this one, which helps streamline further calculations and makes complex-looking expressions more manageable. Understanding and applying trigonometric identities is crucial as they often arise in problems involving trigonometric functions. The identity we used is one of many that can be incredibly useful:

• \(\cos^2 x - \sin^2 x = \cos(2x)\)
• \(\sin^2 x + \cos^2 x = 1\)

Such identities not only simplify our expressions, but also open up pathways for additional insight into the problems we're tackling.

Differentiation Steps Leading to the Second Derivative

Differentiation is a process that allows us to find the rate at which a function is changing. To find the second derivative, we first need to calculate the first derivative, and then differentiate it again. Starting with our function \(y = \sin x \cdot \cos x\), we employed the product rule to find:First derivative:\[\frac{dy}{dx} = \cos^2 x - \sin^2 x = \cos(2x).\]Next, we differentiate \(\cos(2x)\) to find the second derivative. This requires understanding of the chain rule, which modifies the basic derivative calculations because \(2x\) is involved rather than just \(x\). The derivative of \(\cos(u)\) is \(-\sin(u) \cdot u'\), where \(u\) is any function of \(x\). Applying it here gives us:Second derivative:\[\frac{d^2 y}{dx^2} = \frac{d}{dx} \left[ \cos(2x) \right] = -\sin(2x) \cdot 2 = -2\sin(2x).\]By carefully working through these differentiation steps, you gain a comprehensive understanding of how changes in the function's behavior can be interpreted through derivatives. Always ensure to apply the rules methodically to reduce errors, especially in complex expressions.