Problem 30 - Thomas Calculus

Question

Find the derivatives of all orders of the functions. $$ y=\frac{x^{5}}{120} $$

Step-by-Step Solution

Answer

The derivatives are: first \(\frac{1}{24}x^4\), second \(\frac{1}{6}x^3\), third \(\frac{1}{2}x^2\), fourth \(x\), fifth \(1\), and sixth and beyond \(0\).

Step 1: Find the First Derivative

To find the first derivative of the function \(y = \frac{x^5}{120}\), we apply the power rule of differentiation. The power rule states that if \(y = x^n\), then \(\frac{dy}{dx} = nx^{n-1}\). Applying this to our function, we have: \[ \frac{dy}{dx} = \frac{5}{120}x^{5-1} = \frac{1}{24}x^4. \]

Step 2: Find the Second Derivative

Now, we find the second derivative, which is the derivative of the first derivative \(\frac{dy}{dx} = \frac{1}{24}x^4\). Using the power rule again, we get: \[ \frac{d^2y}{dx^2} = \frac{4}{24}x^{4-1} = \frac{1}{6}x^3. \]

Step 3: Find the Third Derivative

To find the third derivative, differentiate \(\frac{d^2y}{dx^2} = \frac{1}{6}x^3\): \[ \frac{d^3y}{dx^3} = \frac{3}{6}x^{3-1} = \frac{1}{2}x^2. \]

Step 4: Find the Fourth Derivative

Find the fourth derivative by differentiating \(\frac{d^3y}{dx^3} = \frac{1}{2}x^2\): \[ \frac{d^4y}{dx^4} = \frac{2}{2}x^{2-1} = x. \]

Step 5: Find the Fifth Derivative

Differentiate \(\frac{d^4y}{dx^4} = x\) to find the fifth derivative: \[ \frac{d^5y}{dx^5} = 1. \]

Step 6: Recognize Higher Order Derivatives

The sixth derivative of the constant \(1\) will be zero, and all higher order derivatives beyond this point will also remain zero. Thus, for any \(n \geq 6\), \(\frac{d^ny}{dx^n} = 0\).

Key Concepts

Power Rule

The Power Rule is a fundamental technique in calculus used to find the derivatives of functions in the form of any power of \( x \). If you have an expression like \( y = x^n \), the Power Rule states that the derivative \( \frac{dy}{dx} \) is given by \( nx^{n-1} \). This means you multiply the whole term by the exponent \( n \) and then subtract one from the exponent. When applying the Power Rule to the function \( y = \frac{x^5}{120} \), it's important to first notice that the function fits the required form, and then proceed with the formula directly.

Higher Order Derivatives

Higher order derivatives involve finding the derivative multiple times. Each subsequent derivative gives insight into the behavior and rate of change of the function beyond just the slope.

• The first derivative \( \frac{dy}{dx} \) can be seen as the rate of change of \( y \) with respect to \( x \).
• The second derivative \( \frac{d^2y}{dx^2} \) provides information on the curvature or concavity of the function.
• Successive derivatives continue to provide more detailed information, such as the rate of change of curvature and so on.

In our solution, we computed up to the fifth derivative. By the time we reached the sixth and beyond, the derivative was zero, indicating that no further changes occur.

Differentiation

Differentiation is the process of finding a derivative, which represents a function's instantaneous rate of change. In calculus, it's essential to master this skill to understand how quantities change over time. Differentiation starts with the basic rules, like the Power Rule, and extends to more complex methods applicable to various functions. When you differentiate a polynomial function, each term can be handled separately using the Power Rule. For \( y = \frac{x^5}{120} \), each step of differentiation reduces the degree of \( x \) by one, gradually simplifying the expression until we find that higher order derivatives produce zero, indicating constant behavior.

Calculus

Calculus is a branch of mathematics focused on changes and motion. It provides tools necessary to model dynamic systems and solve problems involving changing quantities. It consists of two main branches: differential calculus and integral calculus.

• Differential Calculus is concerned with the concept of a derivative, helping us understand how a function changes. Our exercise focused on differentiating, which lies within this field.
• Integral Calculus, on the other hand, deals with accumulation and areas under curves, which is related but separate from our current topic.

Understanding these fields extensively enhances one's ability to apply mathematical concepts to real-world scenarios, making calculus a crucial subject in science and engineering.