Problem 34 - University Calculus: Early Transcendentals

Question

Find \(y^{(4)}=d^{4} y / d x^{4}\) if a. \(y=-2 \sin x\) b. \(y=9 \cos x\)

Step-by-Step Solution

Answer

(a) \(y^{(4)} = -2 \sin x\); (b) \(y^{(4)} = 9 \cos x\)

Step 1: Differentiate the function for the first time (a)

For the function \(y = -2 \sin x\), the first derivative is \(y' = \frac{d(-2 \sin x)}{dx} = -2 \cos x\).

Step 2: Differentiate the function a second time (a)

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

Step 3: Differentiate the function a third time (a)

The third derivative is \(y''' = \frac{d(2 \sin x)}{dx} = 2 \cos x\).

Step 4: Differentiate the function a fourth time (a)

For the fourth derivative, we find \(y^{(4)} = \frac{d(2 \cos x)}{dx} = -2 \sin x\).

Step 1: Differentiate the function for the first time (b)

For the function \(y = 9 \cos x\), the first derivative is \(y' = \frac{d(9 \cos x)}{dx} = -9 \sin x\).

Step 2: Differentiate the function a second time (b)

The second derivative is \(y'' = \frac{d(-9 \sin x)}{dx} = -9 \cos x\).

Step 3: Differentiate the function a third time (b)

The third derivative is \(y''' = \frac{d(-9 \cos x)}{dx} = 9 \sin x\).

Step 4: Differentiate the function a fourth time (b)

The fourth derivative is \(y^{(4)} = \frac{d(9 \sin x)}{dx} = 9 \cos x\).

Key Concepts

Fourth Derivative

In calculus, finding the fourth derivative of a function involves differentiating the function four times. The process of differentiation is about finding the rate at which a function changes. In this particular exercise, we worked on two trigonometric functions:

• For the function \(y = -2 \sin x\), the journey of differentiation takes us back to the original function after four steps. We started with \(-2 \cos x\), then \(2 \sin x\), followed by \(2 \cos x\), and finally returned to \(-2 \sin x\).
• Similarly, for \(y = 9 \cos x\), we moved through \(-9 \sin x\), \(-9 \cos x\), \(9 \sin x\), and ended up back to \(9 \cos x\).

This cyclic nature of derivatives with trigonometric functions is a fascinating aspect of calculus. It confirms that a fourth derivative can bring us back to our original function, which often happens with trigonometric functions due to their periodic nature.
This recurrence happens because the derivatives of sine and cosine functions alternate between each other with a sign change, leading to a predictable pattern.

Trigonometric Functions

Trigonometric functions such as sine and cosine are fundamental in calculus due to their repetitive oscillating nature. These functions have several distinct properties:

• **Sine Function (\(\sin x\))**: When differentiated for the first time, the result is the cosine function. Each subsequent derivative continues a predictable pattern: sine, cosine, negative sine, negative cosine, and so on.
• **Cosine Function (\(\cos x\))**: The cosine function follows a similar pattern, with derivatives flipping between negative and positive while alternating between cosine and sine functions.

These alternating derivatives make trigonometric functions unique, preserving the same sequence in cycles of four. The property that \(\sin x\) and \(\cos x\) derive into each other is invaluable in calculus, especially when working through problems involving oscillations or waves.

Differentiation Steps

The differentiation process involves sequentially finding the derivatives of a function. Let's break down these steps:

• **First Differentiation**: When differentiating \(y = -2 \sin x\), the first derivative is \(-2 \cos x\). For \(y = 9 \cos x\), the first derivative is \(-9 \sin x\). This involves applying basic differentiation rules known as chain rules and derivatives of trigonometric functions.
• **Second Differentiation**: We continue by taking the derivative of the result from the first step. This results in switching the function from cosine to sine, or vice versa, and sometimes introduces a negative sign.
• **Third and Fourth Differentiation**: Continuing this pattern leads us to a third and then finally back to our original expression at the fourth integration.

Understanding this step-by-step method is crucial as it highlights how applying basic formulas repetitively aligns with periodic functions like sine and cosine. This recurring cycle illustrates an essential property of trigonometric differentiation, providing insights that are useful in various areas of mathematics and engineering.