Problem 211 - Calculus Volume 1

Question

For the following exercises, find the requested higher-order derivative for the given functions. $$\frac{d^{4} y}{d x^{4}}\text { of } y=5 \cos x$$

Step-by-Step Solution

Answer

The fourth derivative of \( y = 5 \cos x \) is \( 5 \cos x \).

Step 1: Identify the function and the required derivative

We are given the function \( y = 5 \cos x \) and the task is to find the fourth derivative, \( \frac{d^{4} y}{dx^4} \).

Step 2: Find the first derivative

Differentiate \( y = 5 \cos x \) with respect to \( x \) to find the first derivative:\[ \frac{dy}{dx} = 5 (-\sin x) = -5 \sin x. \]

Step 3: Find the second derivative

Differentiate \( -5 \sin x \) with respect to \( x \) to find the second derivative:\[ \frac{d^{2}y}{dx^2} = -5 (\cos x) = -5 \cos x. \]

Step 4: Find the third derivative

Differentiate \( -5 \cos x \) with respect to \( x \) to find the third derivative:\[ \frac{d^{3}y}{dx^3} = -5 (-\sin x) = 5 \sin x. \]

Step 5: Find the fourth derivative

Differentiate \( 5 \sin x \) with respect to \( x \) to find the fourth derivative:\[ \frac{d^{4}y}{dx^4} = 5 (\cos x) = 5 \cos x. \]

Step 6: Verify pattern and result

Notice the derivatives of \( y = 5 \cos x \) cycle every four differentiations back to \( 5 \cos x \):- \( \cos \):(first derivative) \( -\sin \) (second derivative) \(-\cos\) (third derivative) \( \sin \) (fourth derivative) \( \cos \). Thus, \( \frac{d^{4} y}{dx^4} = 5 \cos x \).

Key Concepts

Derivative Calculation

In calculus, calculating the derivative of a function is a fundamental skill. When we talk about higher-order derivatives, we are differentiating a function multiple times to see how its rate of change evolves. For a given function like \( y = 5\cos(x) \), the task might be to find the fourth derivative, written as \( \frac{d^4y}{dx^4} \). To achieve this, we begin by identifying the function and what we are differentiating it with respect to, in this case, \( x \).

The process involves several consecutive differentiations:

• First, we find the first derivative, which is \( -5\sin(x) \), because the derivative of \( \cos(x) \) is \( -\sin(x) \).
• Next, the second derivative: differentiate again to get \( -5\cos(x) \).
• For the third derivative, differentiate \( -5\cos(x) \) to obtain \( 5\sin(x) \).
• Finally, the fourth derivative brings us back to \( 5\cos(x) \).

After the fourth derivative, we observe a repeating pattern. Understanding how to calculate these derivatives helps reveal the periodic nature of trigonometric functions like sine and cosine.

Trigonometric Functions

Trigonometric functions such as sine and cosine are essential in calculus due to their periodic nature. They describe oscillatory behaviors and circular motion, which are common in physics and engineering.

For our function \( y = 5\cos(x) \), knowing the derivatives of \( \cos(x) \) and \( \sin(x) \) is crucial for correctly computing higher-order derivatives:

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

Each derivative of a cosine or sine function leads to another trigonometric function. This alternating process highlights a unique cyclic nature. Calculating successive derivatives, as shown in the original exercise, demonstrates this cycle:

1. \( \cos(x) \rightarrow -\sin(x) \)
2. \( -\sin(x) \rightarrow -\cos(x) \)
3. \( -\cos(x) \rightarrow \sin(x) \)
4. \( \sin(x) \rightarrow \cos(x) \)

This four-step cycle is important for problems involving repeated derivations of trigonometric functions.

Calculus Problems

Solving calculus problems, especially those involving higher-order derivatives, involves methodical steps and clear understanding of fundamental concepts. Each step in a derivative calculation builds upon the previous, which requires consistency and precision.

One common challenge in calculus problems is keeping track of cycles and patterns, especially with periodic functions such as trigonometric functions. In our example, once you reach the fourth derivative of the function \( y = 5\cos(x) \), you notice it aligns with the original function. This reveals a repeating pattern useful for recognizing solutions quickly in similar problems.

When solving calculus problems like these:

• Start by clearly identifying the function and what derivative you need.
• Work step-by-step, verifying each derivative through differentiation rules.
• Look for patterns or cycles that might simplify your calculations.
• Practice identifying and applying rules correctly to build confidence in complex problems.

Approaching calculus systematically aids in understanding and solving more advanced mathematical problems efficiently.