Problem 26 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are given two different functions for which we need to find the fourth derivative with respect to \(x\). For part (a), we have \(y = -2 \sin x\) and for part (b), \(y = 9 \cos x\).

Step 2: Derivative of Sine Function

For the function \(y = -2 \sin x\), the first derivative is \(y' = -2 \cos x\).

Step 3: Successive Derivatives of Sine Function

From \(y' = -2 \cos x\), differentiate to get \(y'' = 2 \sin x\).Differentiate again to get \(y^{(3)} = 2 \cos x\).Differentiate one more time to get the fourth derivative \(y^{(4)} = -2 \sin x\).

Step 4: Verify Sine Function Fourth Derivative

Observe that taking the fourth derivative of \(y = -2 \sin x\) returns to the original function due to cyclic behavior of sine derivatives: \(y^{(4)} = -2 \sin x\).

Step 5: Derivative of Cosine Function

For \(y = 9 \cos x\), the first derivative is \(y' = -9 \sin x\).

Step 6: Successive Derivatives of Cosine Function

From \(y' = -9 \sin x\), differentiate to get \(y'' = -9 \cos x\).Differentiate again to get \(y^{(3)} = 9 \sin x\).Differentiate one more time to get the fourth derivative \(y^{(4)} = 9 \cos x\).

Step 7: Verify Cosine Function Fourth Derivative

Notice that for the cosine function, the fourth derivative loops back to the original function: \(y^{(4)} = 9 \cos x\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics and are frequently used for modeling periodic phenomena. Two of the basic trigonometric functions are the sine and cosine functions. Trigonometric functions can be visualized as unit circle rotations, where each point on the circle represents an angle and its arc length. These functions are periodic, which means their values repeat at regular intervals, specifically every \(2\pi\) radians. This periodic nature is particularly important when working with derivatives. For instance, finding the fourth derivatives of sine and cosine functions illustrates this periodicity because the functions naturally loop back to their original forms. Understanding this concept helps when calculating higher order derivatives.

Sine Function

The sine function, denoted as \( \sin x \), is a fundamental trigonometric function that represents the y-coordinate of a point on a unit circle. When we differentiate \( \sin x \), we observe a cycle of derivative changes:

• The first derivative of \( \sin x \) is \( \cos x \).
• The second derivative of \( \sin x \) is \( -\sin x \).
• The third derivative of \( \sin x \) is \( -\cos x \).
• The fourth derivative takes us back to \( \sin x \).

This cyclic nature repeats every four derivatives and is due to the periodic behavior of the sine function. For instance, differentiating \(-2 \sin x\) successively brings us back to the original form in the fourth derivative step, \( -2 \sin x \). This implies that higher order derivatives of sine functions are often simpler to compute because of this repeating pattern. Recognizing this cycle can save time in calculations and provides a fundamental insight into the structure of sine-based equations.

Cosine Function

The cosine function, denoted as \( \cos x \), is another primary trigonometric function that represents the x-coordinate of a point on a unit circle. Like the sine function, the cosine function also exhibits a cycle of derivatives:

• The first derivative of \( \cos x \) is \( -\sin x \).
• The second derivative of \( \cos x \) is \( -\cos x \).
• The third derivative of \( \cos x \) is \( \sin x \).
• The fourth derivative returns to \( \cos x \).

This periodic and cyclic nature implies that any fourth derivative of a cosine function will revert to its original function. Taking successive derivatives of functions like \( 9 \cos x \) will similarly repeat through this cycle, returning to \( 9 \cos x \) after four derivatives. This pattern is crucial, as it not only simplifies derivative calculations but also strengthens understanding of the periodic traits inherent in trigonometric functions.