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Derivatives are a fundamental concept in calculus. They measure the rate at which a function changes as its input changes. Simply put, the derivative of a function represents its slope or steepness at any given point. When we talk about finding the derivative of a function with respect to a variable, we're looking at how the function changes as that variable changes.

In this exercise, we aimed to find the derivative of the function \( y = \frac{1}{18}(3x-2)^6 + (4-\frac{1}{2x^2})^{-1} \). To do this, we applied different rules, including the chain rule and the power rule, to differentiate each term in the function. By doing so, we could determine the rate of change of the entire function with respect to \( x \).

Understanding derivatives is crucial because they help us analyze and predict the behavior of functions, which is fundamental in fields such as physics, engineering, and economics.

The chain rule is a powerful tool in calculus used to find the derivative of composite functions. A composite function is one where a function is applied inside another function, like \( f(g(x)) \). The chain rule helps us differentiate such functions by handling the "inside" and "outside" functions separately.

To apply the chain rule, we differentiate the "outside" function first, leaving the inside function unchanged. Then, we multiply that result by the derivative of the "inside" function. This is exactly what we did with the first term of our original function, \( y_1 = \frac{1}{18}(3x-2)^6 \).

:- We let \( u = 3x-2 \). Thus, \( y_1 = \frac{1}{18}u^6 \). - Differentiate the outside function: \( \frac{dy_1}{du} = \frac{1}{18} \cdot 6u^5 \). - Find \( \frac{du}{dx} = 3 \), so multiply: \( \frac{dy_1}{dx} = \frac{1}{18} \cdot 6u^5 \cdot 3 \).

As you can see, the chain rule allows us to simplify the process of finding derivatives of nested functions, like polynomials, exponentials, and trigonometric functions.

The power rule is one of the most straightforward differentiation rules in calculus. It states that if you have a function of the form \( f(x) = x^n \), its derivative is \( f'(x) = n \cdot x^{n-1} \). This rule makes it easy to find the derivative of any function involving powers of \( x \).

In our exercise, the power rule was applied during the differentiation of the term \( \frac{1}{18}(3x-2)^6 \). Here's how it works:

• Identify the power term: \( (3x-2)^6 \).
• Differentiate using the power rule: bring down the exponent (6 in this case), and subtract 1 from the exponent.

Using this method:- We multiplied by the constant: \( \frac{1}{18} \cdot 6 \cdot (3x-2)^5 \). - This gave us a key part of the derivative when combined with the chain rule.

The power rule makes dealing with polynomial functions simple and efficient, avoiding more complex calculations often associated with more intricate rules.