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In calculus, the derivative of a function represents the rate at which the function value changes as its input changes. If we have a function 'y' that depends on 'x', denoted as 'y = f(x)', its derivative is represented as \( \frac{dy}{dx} \) or \( f'(x) \). It tells us how 'y' moves when 'x' is nudged slightly. The derivative can also be understood as the slope of the tangent line to the function at any given point.

For example, if 's(t)' is a function representing the position of an object over time 't', then \( \frac{ds}{dt} \) describes its velocity—how position changes with time. The concept of derivatives is fundamental in physics, engineering, and many other fields that model dynamic systems.

A function is said to be differentiable at a point if its derivative exists at that point. This means that the function has a defined slope at that point, and it behaves nicely - without sharp corners or discontinuities. For a function to be differentiable everywhere, it must have a derivative at every point in its domain.

When working with differentiable functions like 'f', 'g', and 'h' in the context of chain rule, we can smoothly determine the derivative of a composition of these functions. Being differentiable does not only mean having a derivative; it also ensures that the function is continuous, which is vital when we're chaining functions together.

A composite function is a function created by combining two or more functions. If we have functions 'f', 'g', and 'h', a composite function might be 'y = f(g(h(x)))', which is the case in our exercise. Here, 'h' is first applied to 'x', then 'g' is applied to the result of 'h(x)', and finally, 'f' is applied to 'g(h(x))'.

It's like nesting dolls, where each function contains another. Understanding composite functions is essential for mastering the chain rule, as it dictates that we take derivatives in the outer-to-inner function order. This way, each function is differentiated in turn, allowing us to handle more complex scenarios where functions are blended together.

There are several calculus techniques critical for solving different types of problems, and one of the most powerful among these is the chain rule. It's a method used to find the derivative of composite functions. The chain rule can be applied multiple times when dealing with several layers of nested functions, as seen in our exercise.

Other important techniques include the product rule, for multiplying functions; quotient rule, for dividing them; and integration by parts, for integrating products of functions. Mastering each technique is like acquiring a new tool that helps you tackle diverse problems in calculus, ensuring you're well-equipped to deal with the varying challenges that mathematical functions present.