Problem 59 - University Calculus: Early Transcendentals

Question

By computing the first few derivatives and looking for a pattern, find \(d^{999} / d x^{999}(\cos x)\).

Step-by-Step Solution

Answer

The 999th derivative of \(\cos x\) is \(-\cos x\).

Step 1: Find the First Derivative

Start by finding the first derivative of \(\cos x\). The derivative of \(\cos x\) is \(-\sin x\).

Step 2: Find the Second Derivative

Differentiate the result from Step 1, \(-\sin x\), to find the second derivative. The derivative of \(-\sin x\) is \(-\cos x\).

Step 3: Find the Third Derivative

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

Step 4: Find the Fourth Derivative

Differentiate \(\sin x\) to get \(\cos x\) as the fourth derivative.

Step 5: Identify the Pattern

Observe the sequence of derivatives: \(\cos x, -\sin x, -\cos x, \sin x, \cos x, \ldots\). Notice that they cycle every four steps.

Step 6: Determine the Pattern's Position for the 999th Derivative

Since the derivatives repeat every four times, calculate \(999 \mod 4\) to find which derivative corresponds to the 999th. This gives \(999 \mod 4 = 3\).

Step 7: Find the 999th Derivative

From the pattern, the third derivative in the four-derivative cycle is \(-\cos x\). Therefore, \(d^{999} / dx^{999}(\cos x) = -\cos x\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and are used to model periodic phenomena. Among these functions, \(\sin\) and \(\cos\) are the most familiar. These functions depend on an angle measured in radians and exhibit wave-like behavior.
For instance, the cosine function \(\cos(x)\) starts at a maximum value when \(x=0\) and repeats its pattern every \(2\pi\) radians. Similarly, the sine function \(\sin(x)\) starts at zero and has a wave that also repeats every \(2\pi\).
Key Properties:

• The maximum values of \(\cos(x)\) and \(\sin(x)\) are 1, and the minimum values are -1.
• These functions are periodic and continuously cycle through their values.
• They have specific symmetry properties. \(\cos(x)\) is an even function, meaning \(\cos(-x) = \cos(x)\). \(\sin(x)\) is an odd function, so \(\sin(-x) = -\sin(x)\).

Understanding these basics provides a solid foundation for more complex operations, such as differentiation.

Cyclic Patterns

Cyclic patterns in mathematics occur when a certain operation, like differentiation, returns to its original state after a fixed number of steps. In the context of derivatives of trigonometric functions, recognizing these patterns can greatly simplify computations.
For the function \(\cos(x)\), as seen in the exercise solution, the derivatives form a repeating cycle:

• First derivative: \(-\sin(x)\)
• Second derivative: \(-\cos(x)\)
• Third derivative: \(\sin(x)\)
• Fourth derivative: \(\cos(x)\)

This cycle repeats every four derivatives, so every fourth derivative returns to the original function \(\cos(x)\). Identifying these cyclic patterns helps in finding higher-order derivatives without excessive computation.
When tasked with finding, for example, the 999th derivative of \(\cos(x)\), you only need to determine the position within the cycle. In the exercise, \(999 \mod 4 = 3\), matching the position of the third derivative, which is \(-\cos(x)\). Recognizing such patterns saves time and helps ensure accuracy in differentiation tasks.

Differentiation Rules

Differentiation rules are essential tools in calculus used to find the rate at which a function changes. Understanding these rules allows us to tackle a variety of functions beyond simple polynomials, including trigonometric functions.
For trigonometric functions:

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

These derivatives arise from the fundamental properties of trigonometric functions and their interactions with angular values.
Differentiation rules extend to cover more complex scenarios through techniques such as:

• Chain Rule: Useful for differentiating composite functions.
• Product Rule: Allows differentiation of the product of two functions.
• Quotient Rule: Used to differentiate a quotient of two functions.

Each of these rules governs how derivatives are calculated in specific situations and are crucial for handling functions like trigonometric combinations and more advanced calculus problems.