Problem 38 - Calculus and its applications

Question

Find \(y^{\prime \prime}\) $$ \text { For } y=x^{4}, \text { find } d^{4} y / d x^{4} $$

Step-by-Step Solution

Answer

The fourth derivative of \( y = x^4 \) is 24.

Step 1: Understanding the Problem

We need to apply the derivative rules to find the fourth derivative of the function \( y = x^4 \). This requires differentiating the function multiple times.

Step 2: First Derivative

Begin by taking the first derivative of \( y = x^4 \). The power rule in calculus tells us that the derivative of \( x^n \) is \( nx^{n-1} \). So, the first derivative is \( y' = 4x^3 \).

Step 3: Second Derivative

Differentiate the result from the first derivative. Apply the power rule again: \( y'' = \frac{d}{dx}(4x^3) = 12x^2 \).

Step 4: Third Derivative

Take the derivative of the second derivative: \( y''' = \frac{d}{dx}(12x^2) = 24x \).

Step 5: Fourth Derivative

Finally, differentiate the third derivative to find the fourth derivative: \( y^{(4)} = \frac{d}{dx}(24x) = 24 \).

Key Concepts

Power Rule

The Power Rule is a fundamental tool in calculus used to find the derivative of functions of the form \( x^n \). It's straightforward and powerful, making differentiation much easier when dealing with polynomial expressions.

• The rule states that if you have a function \( y = x^n \), the derivative \( y' \) is given by \( nx^{n-1} \).
• For instance, if you need to differentiate \( y = x^4 \), you apply the Power Rule to get \( y' = 4x^3 \).

This rule holds true for any real number \( n \), simplifying the process of finding derivatives when applied repeatedly, as seen when calculating higher-order derivatives.

Differentiation

Differentiation is the process of finding the derivative of a function. It essentially measures how a function changes as the input changes. This

• Allows us to determine rates of change, such as velocity when differentiating position with respect to time.
• Enables the calculation of slopes of tangent lines to curves, critical for understanding function behavior.

In our exercise, you apply differentiation multiple times: each time you differentiate, you reduce the degree of the polynomial by one, following the Power Rule.

• First, find \( y' = 4x^3 \).
• Then, for \( y'' = \frac{d}{dx}(4x^3) = 12x^2 \).
• Next, go to \( y''' = \frac{d}{dx}(12x^2) = 24x \).
• Finally, the fourth derivative becomes \( y^{(4)} = \frac{d}{dx}(24x) = 24 \).

Fourth Derivative

The fourth derivative of a function is a higher-order derivative obtained by differentiating the third derivative. This process is part of analyzing how a function's rate of change itself changes, and it becomes quite crucial in advanced calculus and real-world applications.As you continue differentiating polynomial functions, notice:

• The power of \( x \) reduces by one each time you differentiate.
• In this specific function \( y = x^4 \), the fourth derivative is eventually a constant: \( y^{(4)} = 24 \).
• This means no matter what \( x \) value you plug into this function, the rate of increase is unchanging.

The process of finding the fourth derivative highlights the cyclical nature of differentiating polynomial functions as their higher derivatives eventually become zero or a constant.