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Magazine Chapter 3: Problem 50
Find the \(50^{\text {th }}\) derivative of \(y=\cos x\).
Short Answer
Expert verified
The \(50^{th}\) derivative of \(y = \cos x\) is \(-\cos x\).
Step by step solution
01
Identify the Problem
We need to find the \(50^{th}\) derivative of the function \(y = \cos x\). This requires understanding the pattern in the derivatives of \( \cos x \).
02
Compute Initial Derivatives
The first few derivatives of \(y = \cos x\) are:\[ \begin{align*}&y' = -\sin x, \&y'' = -\cos x, \&y''' = \sin x, \&y^{(4)} = \cos x. \\end{align*} \]
03
Determine the Pattern
Notice that the derivatives repeat every four derivatives: \( \cos x, -\sin x, -\cos x, \sin x, \cos x, \ldots \). This means the cycle of derivatives has a period of 4.
04
Identify the Cycle Position
To find the \(50^{th}\) derivative, find the position within the cycle of 4. Compute \(50 \mod 4\), which gives a remainder of 2, indicating that the \(50^{th}\) derivative is the same as the \(2^{nd}\) derivative.
05
Determine the Specific Derivative
From the earlier results, the \(2^{nd}\) derivative of \(y = \cos x\) is \(-\cos x\). Therefore, \(y^{(50)} = -\cos x\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Cosine Function
The cosine function, denoted as \( \cos x \), is one of the primary trigonometric functions that you'll encounter. It describes a wave-like pattern that cycles through its values from 1 to -1 and back. This function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) units.
- The cosine function takes an angle as its input and returns a value between -1 and 1.
- It's closely related to the other main trigonometric functions: sine \( \sin x \) and tangent \( \tan x \).
- The graph of \( \cos x \) starts at 1 when \( x = 0 \), decreases to -1, and returns to 1 completing its cycle at \( x = 2\pi \).
Exploring the Derivative Cycle
In calculus, the derivative of a function measures how the function's output changes as its input changes. For periodic functions like the cosine function, these changes have a cyclic nature. When you compute derivatives of \( \cos x \), a fascinating pattern arises.
- The first derivative of \( \cos x \) is \( -\sin x \).
- Continuing, the second derivative is \( -\cos x \).
- The third is \( \sin x \), and the fourth returns to \( \cos x \).
Using the Modulus Operation
The modulus operation is a mathematical concept used to find the remainder of a division between two numbers. It is crucial when determining cycle positions in repeating patterns, such as the derivatives of a trigonometric function.
- In the context of finding the \(50^{th}\) derivative of \( \cos x \), we determined the repetitive cycle length is 4.
- Using the modulus operation \(50 \mod 4\), we find the remainder is 2.
- This result tells us that the \(50^{th}\) derivative corresponds to the same position in the cycle as the \(2^{nd}\) derivative.
An Overview of Trigonometric Functions
Trigonometric functions are fundamental in mathematics, particularly in calculus and geometry. They relate the angles of a triangle to the ratios of its sides. The core set includes sine, cosine, and tangent, each with its own unique properties and applications.
- \( \cos x \) is the adjacent side over hypotenuse in a right-angled triangle.
- \( \sin x \) represents the opposite side over hypotenuse.
- \( \tan x \) is the ratio \( \sin x / \cos x \).
