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The chain rule is a fundamental tool in calculus for computing the derivative of a composite function. For example, if we have a function that is composed of two other functions, say \( u(x) \) and \( v(x) \), where \( u = g(v(x)) \), the derivative of \( u \) with respect to \( x \) can be found by differentiating \( u \) with respect to \( v \) and then multiplying it by the derivative of \( v \) with respect to \( x \).

The general formula for the chain rule looks like this: \[\frac{du}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx}\]. In the exercise given, the chain rule is applied to the function \( h(x) \) to find its derivative. This step is crucial because understanding the derivative of \( h(x) \) helps in analyzing the increasing and decreasing intervals of the function. Furthermore, when applied correctly, the chain rule simplifies the process of finding derivatives for complex functions and saves time when solving calculus problems.

A derivative, in the context of differential calculus, is a way to show the rate at which a function is changing at any given point. Technically, it's the limit of the difference quotient as the difference in \(x\) approaches zero, often interpreted as the slope of the tangent line to the function's graph at a particular point.

For a function \( f(x) \), the derivative will be denoted as \( f'(x) \), and if \( f'(x) > 0 \) for all \( x \) in a certain interval, the function is said to be increasing on that interval. Conversely, if \( f'(x) < 0 \) for all \( x \) on an interval, the function is decreasing there. Understanding this concept is essential when looking to find where the function \( h(x) \) increases or decreases. The derivative tells the story of the function's growth and decay, a story that unfolds through its values and signs across different intervals.

Increasing and decreasing intervals refer to the portions of a function's domain where the function is consistently moving upwards or downwards, respectively. To determine these intervals for a differentiable function, one must look at the sign of the function's derivative.

When we have \( f'(x) > 0 \) for all \( x \) within an interval, the function \( f(x) \) is increasing on that interval. Alternatively, if \( f'(x) < 0 \) on an interval, then \( f(x) \) is decreasing there. These intervals provide key insights into the behavior of the function. As shown in the step-by-step solution, without more specific information about the functions \( f(x) \) and their relationship, it's challenging to determine precisely where the composite function \( h(x) \) is increasing or decreasing. This limitation highlights the necessity to analyze the function's derivative closely and understand its implications on the function's growth trends.