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Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. In simpler terms, it is used to find the derivative, which gives us a mathematical description of how a function's output value changes as its input value changes. Differentiation is essential for analyzing graphs, determining slopes of curves, and solving real-world problems such as motion and growth.

Key aspects of differentiation include:

• Identifying the function you want to differentiate.
• Applying the correct differentiation rules, such as product rule, quotient rule, or chain rule, based on the function's form.
• Understanding that differentiation requires careful handling of algebraic expressions to achieve accurate results.

In the given exercise, differentiation is used to identify the change rate of the function \((3x-9)^{-5/3}\). This requires understanding the nested nature of functions and applying the chain rule to manage this complexity.

Calculus is a branch of mathematics that studies continuous change. It is divided into two main areas: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which measures how a quantity changes in response to changes in another quantity.

In practical terms, calculus allows us to describe how things change over time and space. It's widely used in a variety of fields such as physics, engineering, economics, and biology. To successfully apply calculus, one needs a strong understanding of functions, limits, and an ability to perform algebraic manipulations.

The given exercise focuses on using differential calculus to determine the rate of change of a power function, involving careful differentiation through techniques like the chain rule.

The derivative of a power function is an expression that shows how a function involving exponents changes as its input changes. A power function is typically of the form \(u(x)^n\), where \(u(x)\) is a differentiable function and \(n\) is a constant exponent. The work for finding the derivative of a power function is entrenched in the rules of differentiation, particularly when you deal with the Chain Rule.

For the derivative of \(u(x)^n\):

• Apply the power rule: Bring down the exponent \(n\) and reduce the original exponent by one, i.e., \(n(u(x))^{n-1}\).
• Multiply this result by the derivative of the inside function \(u(x)\).

In our exercise, the function \(y = (3x-9)^{-5/3}\) requires these steps, identifying \(u(x) = 3x - 9\) and using the power rule combined with the chain rule to find the derivative correctly. As a result, you can derive the pattern that shows how the function's gradient alters at different points.