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Hyperbolic functions are analogues of the ordinary trigonometric, or circular, functions. Just as trigonometric functions are related to the circle, hyperbolic functions are associated with a hyperbola. Two fundamental hyperbolic functions are the hyperbolic sine and cosine functions, denoted as \( \sinh(x) \) and \( \cosh(x) \) respectively.

The definitions of these functions using exponential functions are \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) and \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). They are useful when modeling certain types of waves and oscillations and appear naturally in the solutions of several differential equations.

Like trigonometric functions, you can construct other hyperbolic functions such as the hyperbolic tangent or hyperbolic cotangent based on \( \sinh \) and \( \cosh \). Importantly, these functions possess properties and identities similar to their trigonometric cousins, like the hyperbolic identity \( \cosh^2(x) - \sinh^2(x) = 1 \), which is analogous to the Pythagorean identity.

In calculus, taking derivatives is a fundamental operation that measures how a function changes as its input changes. The derivative of the hyperbolic sine function, \( \sinh(x) \), is significant because it helps in solving problems involving hyperbolic functions. The beauty of hyperbolic functions is shown in their derivatives; remarkably, the derivative of \( \sinh(x) \) is the hyperbolic cosine function, \( \cosh(x) \).

To put it mathematically, \( \frac{d}{dx}\sinh(x) = \cosh(x) \). This derivation comes from applying the definition of \( \sinh \), using the rules of differentiation on exponential functions, and simplifying. Knowing this derivative is critical when dealing with more complex functions involving \( \sinh \), where the chain rule might also be applied.

The chain rule is a powerful tool in differential calculus, used when differentiating composite functions. When a function is composed of an 'inner' function and an 'outer' function, the chain rule guides us in breaking down the process of differentiation into manageable steps.

In mathematical terms, if \( f(x) = g(h(x)) \), then the derivative of \( f \) with respect to \( x \) is \( f'(x) = g'(h(x)) \cdot h'(x) \). This means you take the derivative of the outer function \( g \) evaluated at the inner function \( h(x) \) and multiply it by the derivative of \( h(x) \).

Applying it to the given exercise where \( f(x) = \sinh(3x) \), we identify \( \sinh(x) \) as the 'outer' function and \( 3x \) as the 'inner' function. Differentiating \( \sinh(x) \) with respect to its argument gives \( \cosh(x) \), and the derivative of the 'inner' \( 3x \) is simply 3. Hence, the chain rule provides us with \( f'(x) = \cosh(3x) \times 3 \), an elegant expression showcasing the harmonious collaboration between hyperbolic function derivatives and the chain rule.