91Ó°ÊÓ

When we discuss derivatives, we're talking about the rate at which a function changes as its input changes. This concept is central to calculus and helps us understand the idea of differentiation. Take, for example, our function, which can be expressed as \(y = (4 - 3x)^9\). Here, our goal is to determine how \(y\) changes as \(x\) changes. This is expressed as the derivative \(\frac{dy}{dx}\).
Derivatives give us insights into various important features of a function:

• They inform us about the slope of a function at any given point, which can indicate how steep or flat the curve is at that point.
• They help in understanding the behavior and trends of functions over an interval, like maxima and minima.
• They are central in forming other complex calculus concepts, such as integrals and differential equations.

In finding the derivative, especially for composite functions like the one we're working with, we use specific rules, such as the chain rule.

Function composition involves creating a new function by combining two functions. The output of one function becomes the input of another. In calculus, this concept is highly valuable when dealing with complex functions.
For example, in our exercise, we express the function \(y = (4 - 3x)^9\) as a composition of two simpler functions:

• \(u = g(x) = 4 - 3x\)
• \(y = f(u) = u^9\)

Here, \(g(x)\) alters the variable \(x\) into an intermediate variable \(u\), which is then transformed by \(f(u)\) into the final form of the function \(y\). By breaking it down, we make complex differentiation tasks more manageable. Understanding function composition helps in visualizing how changes in \(x\) affect \(y\) through intermediate stages, making it easier to apply differentiation rules like the chain rule.

The power rule is a basic yet powerful tool in calculus used to find the derivative of functions of the form \(y = x^n\). According to this rule, if \(y = x^n\), then the derivative \(\frac{dy}{dx} = nx^{n-1}\).
When we encounter functions where variables are raised to a power, the power rule quickly gives us the derivative. This is especially clear in our example with \(u^9\), where applying the power rule gives \(f'(u) = 9u^8\).
Using the power rule involves:

• Multiplying the function by the power (in our case, \(9\)).
• Decreasing the power by one (which turns \(u^9\) into \(9u^8\)).

The power rule simplifies differentiation immensely and is often one of the first rules taught due to its broad applicability. It's a quick path to finding derivatives without needing to work through more involved limit processes.