91Ó°ÊÓ

In calculus, the concept of a derivative is central. It expresses how a function changes as its input changes. A function derivative is essentially a formula that gives us the instantaneous rate of change of the function with respect to its input variable. For instance, if we have a function \( f(x) \), its derivative, denoted as \( \frac{d}{dx}[f(x)] \), tells us how \( f(x) \) grows or shrinks as \( x \) changes.

• A positive derivative indicates that the function is increasing.
• A negative derivative suggests the function is decreasing.
• A zero derivative shows the function may have reached a maximum or minimum point.

Understanding the role of the derivative helps solve more complex problems in calculus. In the given exercise, the derivative of \( f(3x) \) with respect to \( x \) is \( 6x \), which we assess using rules of differentiation, specifically the chain rule to find \( \frac{d}{dx}[f(x)] \). The task is to simplify it back to the basic \( x \) variable.

Differentiation techniques are methods applied to find the derivative of functions. Among these, one of the most versatile is the **Chain Rule**. This rule is crucial when dealing with nested functions, where one function is inside another, like in \( f(g(x)) \). The chain rule can be summarized as:

• If \( y = f(u) \) and \( u = g(x) \), then \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \).

In our worked example, we identify \( u = 3x \), thus \( \frac{du}{dx} = 3 \). Knowing \( \frac{d}{dx}[f(3x)] = 6x \), the chain rule allows us to decompose this into two parts: \( \frac{df}{du} \times \frac{du}{dx} = 6x \).

This reveals that \( \frac{df}{du} = 2x \), which assists us in stepping into finding the final required expression for \( \frac{d}{dx}[f(x)] \). Exploring such techniques ensures mastery over calculus problems.

Solving calculus problems involves more than applying formulas; it requires a methodical approach for accuracy. When tackling a calculus problem:

• First, clearly understand the problem and the function given.
• Identify differentiation rules that apply, such as the chain rule.
• Execute the rules step-by-step, as we did to transform \( \frac{d}{dx}[f(3x)] = 6x \) to \( \frac{d}{dx}[f(x)] = \frac{2}{3}x \).

In our exercise, the primary challenge was in reversing engineered steps from \( f(3x) \) back to \( f(x) \).

This involves a thorough understanding of substitution methods, allowing one to switch back from \( u = 3x \) to \( x = \frac{u}{3} \). Every substitution or calculation must be accurately interpreted to ensure the final expression obtained reflects the correct derivative with respect to \( x \). By applying sound problem-solving strategies within calculus, one can tackle seemingly complex problems with clarity and precision.