91Ó°ÊÓ

Let's dive into the power rule of differentiation. It's one of the simplest rules but immensely powerful. It states that if you have a function in the form of a power, like \(x^n\), then its derivative is \(nx^{n-1}\). This is very straightforward but crucial for understanding how differentiation works. For example, if you have \(x^{7/2}\), according to the power rule, the derivative is obtained by bringing the exponent in front and then reducing the exponent by 1: \[ \frac{d}{dx} x^{7/2} = \frac{7}{2} x^{(7/2) - 1} = \frac{7}{2} x^{5/2} \] This rule simplifies differentiation and makes it easier to handle functions with power terms. Notice how seamless it is when applied to more complicated functions with different exponents.

Now, let’s talk about higher-order derivatives. These are derivatives of derivatives. In other words, you differentiate a function once to get the first derivative, and you can differentiate the result again to get the second derivative, and so on. This process continues until you reach the desired order. In the given exercise, you are asked to find the fourth derivative, denoted as \(D_{x}^{4} y\). To accomplish this, follow these steps:

• First, find the first derivative \( \frac{d}{dx} y \) as we did in the steps outlined in the solution.
• Then, differentiate the first derivative to obtain the second derivative.
• Repeat the process to get the third and fourth derivatives.

Each step builds upon the previous derivative, relying heavily on the ability to consistently apply the differentiation rules, such as the power rule mentioned earlier.

Term-by-term differentiation is simply applying the differentiation rules to each term of a sum or difference separately. In the exercise, the function \( y = x^{7/2} - 2x^{5/2} + x^{1/2} \) can be split and differentiated individually. Here’s how you can do it:

• First, isolate each term: \[ y_1 = x^{7/2} \], \[ y_2 = -2x^{5/2} \], and \[ y_3 = x^{1/2} \]
• Then, differentiate each one using the power rule: \[ \frac{d}{dx} y_1 = \frac{7}{2} x^{5/2} \], \[ \frac{d}{dx} y_2 = -5x^{3/2} \], and \[ \frac{d}{dx} y_3 = \frac{1}{2} x^{-1/2} \]
• Finally, combine these results to form the complete derivative: \[ \frac{d}{dx} y = \frac{7}{2} x^{5/2} - 5 x^{3/2} + \frac{1}{2} x^{-1/2} \]

Term-by-term differentiation makes complex functions easier to manage by breaking them down. This technique is extremely useful for functions with multiple terms.