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Chapter 3: Problem 59

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

Short Answer

Expert verified
The 999th derivative of \(\cos x\) is \(\sin x\).

Step by step solution

01

Find the First Derivative

To find the first derivative of \(\cos x\), we use the basic differentiation rule for trigonometric functions. The derivative of \(\cos x\) is \(-\sin x\). Therefore: \[\frac{d}{dx} \cos x = -\sin x\]
02

Find the Second Derivative

Taking the derivative of the first derivative (\(-\sin x\)), we get the second derivative as the derivative of \(-\sin x\), which is \(-\cos x\). Therefore: \[\frac{d^2}{dx^2} \cos x = -\cos x\]
03

Find the Third Derivative

Now, differentiate the second derivative \(-\cos x\). The derivative of \(-\cos x\) is \(\sin x\). Therefore: \[\frac{d^3}{dx^3} \cos x = \sin x\]
04

Find the Fourth Derivative

Differentiate the third derivative \(\sin x\) to find the fourth derivative. The derivative of \(\sin x\) is \(\cos x\). Therefore: \[\frac{d^4}{dx^4} \cos x = \cos x\]
05

Recognize the Pattern

We have observed the derivatives cycle every four terms: - 1st derivative: \(-\sin x\)- 2nd derivative: \(-\cos x\)- 3rd derivative: \(\sin x\)- 4th derivative: \(\cos x\) This shows a repeating cycle of length 4: \(\cos x, -\sin x, -\cos x, \sin x\).
06

Find the 999th Derivative

The derivatives repeat every four terms, so to find the 999th derivative, determine the cycle position within the repetition. Calculate \(999 \mod 4 = 3\) (remainder 3). This means the 999th derivative corresponds to the third position in the cycle, which is \(\sin x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions are mathematical functions based on angles, often used to describe periodic phenomena. The most common trigonometric functions are sine (\(\sin x\)) and cosine (\(\cos x\)). These functions have important properties that are key in calculus and can often be found as solutions to various types of differential equations. In the context of differentiation, trigonometric functions have specific derivatives: the derivative of \(\cos x\)is \(-\sin x\), and the derivative of \(\sin x\)is \(\cos x\). These derivatives are fundamental to understanding more complex mathematical models, especially those involving periodic processes like sound waves or circular motion. Understanding trigonometric functions' derivatives helps us to analyze and predict these models accurately.
Derivative Patterns
When differentiating functions, certain patterns often emerge. These patterns simplify the process and are invaluable in solving higher-order derivatives. For trigonometric functions, these patterns are particularly clear. Take the example of differentiating \(\cos x\):
  • First derivative: \(-\sin x\)
  • Second derivative: \(-\cos x\)
  • Third derivative: \(\sin x\)
  • Fourth derivative: \(\cos x\)
From the sequence above, a recurring cycle every four derivatives appears. Recognizing such patterns helps quickly find higher-order derivatives without repetitive calculations. This efficiency is crucial in contexts like physics and engineering, where higher-order derivatives can describe acceleration and other complex phenomena.
Cycle of Derivatives
The cycle of derivatives is a fascinating characteristic of differentiating functions multiple times. This cycle is evident when working with trigonometric functions like \(\cos x\)or \(\sin x\). For instance, when differentiating \(\cos x\), the derivatives repeat every four steps. This cycle is:
  • \(\frac{d}{dx} \cos x = -\sin x\)
  • \(\frac{d^2}{dx^2} \cos x = -\cos x\)
  • \(\frac{d^3}{dx^3} \cos x = \sin x\)
  • \(\frac{d^4}{dx^4} \cos x = \cos x\)
Knowing that trigonometric functions repeat in this manner, you can tackle problems that require very high-order derivatives. For example, determining the 999th derivative of \(\cos x\). By recognizing the cycle's length, you calculate \(999 \mod 4 = 3\), identifying the cycle's position and arriving at the solution \(\sin x\). Grasping this cycle makes complex calculations much simpler, saving time and effort.

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Most popular questions from this chapter

Find \(d y / d x\) if \(y=x^{3 / 2}\) by using the Chain Rule with \(y\) as a composite of $$\begin{array}{l}{\text { a. } y=u^{3} \text { and } u=\sqrt{x}} \\ {\text { b. } y=\sqrt{u} \text { and } u=x^{3}}\end{array}$$

Use a CAS to perform the following steps for the functions \begin{equation} \begin{array}{l}{\text { a. Plot } y=f(x) \text { to see that function's global behavior. }} \\ {\text { b. Define the difference quotient } q \text { at a general point } x, \text { with }} \\ {\text { general step size } h .} \\\ {\text { c. Take the limit as } h \rightarrow 0 . \text { What formula does this give? }} \\ {\text { d. Substitute the value } x=x_{0} \text { and plot the function } y=f(x)} \\ \quad {\text { together with its tangent line at that point. }} \\ {\text { e. Substitute various values for } x \text { larger and smaller than } x_{0} \text { into }} \\ \quad {\text { the formula obtained in part (c). Do the numbers make sense }} \\ \quad {\text { with your picture? }} \\ {\text { f. Graph the formula obtained in part (c). What does it mean }} \\ \quad {\text { when its values are negative? Zero? Positive? Does this make }} \\ \quad {\text { sense with your plot from part (a)? Give reasons for your }} \\ \quad {\text { answer. }}\end{array} \end{equation} $$f(x)=x^{2} \cos x, \quad x_{0}=\pi / 4$$

Use a CAS to perform the following steps for the functions \begin{equation} \begin{array}{l}{\text { a. Plot } y=f(x) \text { to see that function's global behavior. }} \\ {\text { b. Define the difference quotient } q \text { at a general point } x, \text { with }} \\ {\text { general step size } h .} \\\ {\text { c. Take the limit as } h \rightarrow 0 . \text { What formula does this give? }} \\ {\text { d. Substitute the value } x=x_{0} \text { and plot the function } y=f(x)} \\ \quad {\text { together with its tangent line at that point. }} \\ {\text { e. Substitute various values for } x \text { larger and smaller than } x_{0} \text { into }} \\ \quad {\text { the formula obtained in part (c). Do the numbers make sense }} \\ \quad {\text { with your picture? }} \\ {\text { f. Graph the formula obtained in part (c). What does it mean }} \\ \quad {\text { when its values are negative? Zero? Positive? Does this make }} \\ \quad {\text { sense with your plot from part (a)? Give reasons for your }} \\ \quad {\text { answer. }}\end{array} \end{equation} $$f(x)=x^{1 / 3}+x^{2 / 3}, \quad x_{0}=1$$

Area The area \(A\) of a triangle with sides of lengths \(a\) and \(b\) enclosing an angle of measure \(\theta\) is $$A=\frac{1}{2} a b \sin \theta.$$ a. How is \(d A / d t\) related to \(d \theta / d t\) if \(a\) and \(b\) are constant? b. How is \(d A / d t\) related to \(d \theta / d t\) and \(d a / d t\) if only \(b\) is constant? c. How is \(d A / d t\) related to \(d \theta / d t, d a / d t,\) and \(d b / d t\) if none of \(a, b,\) and \(\theta\) are constant?

Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period \(T\) and the length \(L\) of a simple pendulum with the equation $$T=2 \pi \sqrt{\frac{L}{g}}$$ where \(g\) is the constant acceleration of gravity at the pendulum's location. If we measure \(g\) in centimeters per second squared, we measure \(L\) in centimeters and \(T\) in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to \(L .\) In symbols, with \(u\) being temperature and \(k\) the proportionality constant, $$\frac{d L}{d u}=k L$$ Assuming this to be the case, show that the rate at which the period changes with respect to temperature is \(k T / 2 .\)
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