Problem 39 - Calculus Early Transcendentals

Question

Make a conjecture about the derivative by calculating the first few derivatives and observing the resulting pattern. $$ \frac{d^{87}}{d x^{87}}[\sin x] $$

Step-by-Step Solution

Answer

The 87th derivative of \( \sin x \) is \(-\cos x \).

Step 1: Understand the Derivative of Sine

The derivative of \( \sin x \) is \( \cos x \). This is a fundamental result in calculus.

Step 2: Calculate the Second Derivative

The second derivative of \( \sin x \) is the derivative of \( \cos x \), which is \(-\sin x \).

Step 3: Calculate the Third Derivative

The third derivative of \( \sin x \) is the derivative of \(-\sin x \), which is \(-\cos x \).

Step 4: Calculate the Fourth Derivative

The fourth derivative of \( \sin x \) is the derivative of \(-\cos x \), which returns to \( \sin x \).

Step 5: Identify the Pattern in Derivatives

From the calculations, we can see a repeating cycle: \( \sin x, \cos x, -\sin x, -\cos x \) which repeats every four derivatives.

Step 6: Determine the 87th Derivative

Since the cycle repeats every 4 derivatives, we calculate \( 87 \mod 4 \), which gives 3. Thus, the 87th derivative corresponds to the third position in the cycle, which is \(-\cos x \).

Key Concepts

Pattern Recognition

Recognizing patterns is a key method when working with derivatives of trigonometric functions. Once you start calculating the derivatives yourself, you'll notice hidden sequences in the results. For example,

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

After these four calculations, you see the pattern cycle repeating itself. The ability to recognize such cycles or sequences when calculating multiple derivatives will save you a lot of time and effort, especially as the number of derivatives grows larger.
To apply patterns, you can use modular arithmetic. In this task, calculating \( 87 \mod 4 \) helps determine the 87th derivative of \( \sin x \). Thus, to find the derivative, you only need to determine which position within the cycle it falls into.

Sine Function

The sine function, \( \sin x \), is a foundational part of trigonometric functions in calculus. Understanding its properties is crucial when delving into derivatives.
As a periodic function, \( \sin x \) has a specific wave-like behavior, with cycles that repeat every \( 2\pi \). This periodic nature is mirrored in its derivatives.
Whenever you take the derivative of \( \sin x \), the resulting function reflects its cyclical nature.

• First comes \( \cos x \), which represents the slope of the sine wave at various points.
• The second derivative, \( -\sin x \), indicates inversion compared to the original \( \sin x \).

The alternating pattern of sign and function type points to deep relationships between sine and its co-functional counterpart, cosine, within trigonometry and calculus.
Understanding how these derivatives work together can demystify their applications in physics, engineering, and other mathematics areas.

Cycle of Derivatives

The "cycle" of derivatives as it applies to trigonometric functions is a central concept. When dealing with functions like \( \sin x \), you might find it surprising how neatly the derivatives follow a repeating cycle. This cycle consists of four steps:

• \( \sin x \)
• \( \cos x \)
• \( -\sin x \)
• \( -\cos x \)

After these four steps, the derivatives return to the original function. This is essential because it informs how we calculate higher-order derivatives.
For example, when asked for the 87th derivative of \( \sin x \), rather than computing all 87 individually, you calculate \( 87 \mod 4 \). The remainder tells you which function in the cycle corresponds to the 87th derivative. Here it is 3, indicating \( -\cos x \).
This cyclical nature provides a powerful shortcut for solving seemingly complex problems, making derivative calculation more efficient and intuitive.