Problem 211 - Calculus Volume 1

Question

Find the requested higher-order derivative for the given functions. $$ \frac{d^{4} y}{d x^{4}} \text { of } y=5 \cos x $$

Step-by-Step Solution

Answer

The fourth derivative is \( 5 \cos x \).

Step 1: Differentiate Once

Compute the first derivative of the function \(y = 5 \cos x\). The derivative of \(\cos x\) is \(-\sin x\), so the first derivative is: \( \frac{dy}{dx} = -5 \sin x \).

Step 2: Differentiate Twice

Differentiate the first derivative \( \frac{dy}{dx} = -5 \sin x \). The derivative of \( -\sin x\) is \(-\cos x\), so the second derivative is: \( \frac{d^{2}y}{dx^{2}} = -5 (-\cos x) = 5 \cos x \).

Step 3: Differentiate Three Times

Differentiate the second derivative \( \frac{d^{2}y}{dx^{2}} = 5 \cos x \). The derivative of \(\cos x\) is \(-\sin x\), so the third derivative is: \( \frac{d^{3}y}{dx^{3}} = -5 \sin x \).

Step 4: Differentiate Four Times

Differentiate the third derivative \( \frac{d^{3}y}{dx^{3}} = -5 \sin x \). The derivative of \(-\sin x\) is \(-\cos x\), so the fourth derivative is \( \frac{d^{4}y}{dx^{4}} = -5 (-\cos x) = 5 \cos x \).

Step 5: Check for Pattern

Notice that the function returns to its original form after every two derivatives. This pattern indicates that continuing the differentiation would indeed result in the same sequence, confirming that \( \frac{d^{4}y}{dx^{4}} = 5 \cos x \).

Key Concepts

Differentiation

Differentiation is one of the core concepts in calculus. It is the process of determining the derivative of a function. The derivative provides us with the rate at which a function is changing at any given point.

To differentiate a function, follow these steps:

• Identify the function you need to differentiate.
• Use the appropriate differentiation rules, such as the power rule, product rule, quotient rule, or chain rule.
• Apply the rules step by step to find the derivative.

In the original exercise, we started with the function \(y = 5\cos x\). Given that the derivative of \(\cos x\) is \(-\sin x\), the first derivative was calculated as \(\frac{dy}{dx} = -5\sin x\). Understanding differentiation is crucial as it forms the basis for solving more complex mathematical and engineering problems.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in calculus. They describe relationships between the angles and sides of a triangle. These functions have specific derivatives that help in evaluating changes in cycles, such as sound waves or light.

In our example, we focus on the trigonometric function \(\cos x\), whose derivatives follow a predictable pattern:

• The derivative of \(\cos x\) is \(-\sin x\).
• The derivative of \(-\sin x\) is \(-\cos x\).
• The derivative of \(-\cos x\) is \(\sin x\).
• The derivative of \(\sin x\) is \(\cos x\).

By observing these patterns, we notice that cycles repeat every four derivatives. In our exercise, this cyclic pattern allows for easier computation of higher-order derivatives.

Calculus Problems

Calculus problems often involve finding higher-order derivatives. These derivatives tell us about the curvature or concavity of graphs, and are used in physics to examine motion, in engineering for signal processing, and in economics for cost functions.

The original problem required the fourth derivative of the function \(y=5\cos x\). By differentiating step by step, we observe a repeating sequence in the derivatives that informs us how the function behaves at different orders. Higher-order derivatives can emphasize certain behaviors or patterns not immediately obvious in the original function.

Tackling calculus problems, especially those involving trigonometric functions, often requires looking for these inherent patterns. Recognizing patterns not only simplifies problems but also deepens understanding of the underlying mathematical principles.