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Hyperbolic functions are analogous to trigonometric functions but are derived from exponential functions. They are very useful in many areas of engineering and mathematics. The hyperbolic tangent function, denoted as \(\tanh x\), is one of the six hyperbolic functions. It is defined as the ratio of the hyperbolic sine function \(\sinh x\) to the hyperbolic cosine function \(\cosh x\), which is given by:
\[\tanh x = \frac{\sinh x}{\cosh x} = \frac{\frac{e^x-e^{-x}}{2}}{\frac{e^x+e^{-x}}{2}} = \frac{e^x-e^{-x}}{e^x+e^{-x}}\]
The hyperbolic sine and cosine functions themselves are defined using exponential functions:

• \(\sinh x = \frac{e^x - e^{-x}}{2}\)
• \(\cosh x = \frac{e^x + e^{-x}}{2}\)

Using these relationships, hyperbolic functions allow us to transform complex expressions involving exponential functions into simpler forms.

Exponential functions have the form \(f(x) = a^x\), where \(a\) is a positive constant. They play a crucial role in many mathematical formulations. In the context of hyperbolic functions, the base of the exponential function is typically \(e\), the natural base, leading to expressions such as \(e^x\).
Exponential functions have unique properties that simplify mathematical calculations. For instance:

• Products: \(e^x \times e^y = e^{x+y}\)
• Quotients: \(\frac{e^x}{e^y} = e^{x-y}\)
• Inverses: \(e^{-x} = \frac{1}{e^x}\)

These properties make exponential functions extremely powerful for solving exponential growth or decay problems. They also underlie the operations performed in the step-by-step solution by allowing \(\tanh x\) to be expressed effectively in terms of exponential functions.

A mathematical proof is a logically valid argument that establishes the truth of a statement beyond any doubt. It involves a series of logical deductions made from axioms or previously established theorems. In our example, we aim to prove that the expression \(\frac{1+\tanh x}{1-\tanh x}\) equals \(e^{2x}\).
The process starts with expressing \(\tanh x\) in terms of exponential functions. By doing so, it lays the groundwork for simplifying the given expression. The proof progresses with logical algebraic manipulations:

• Substitute \(\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}\) into the target expression.
• Simplify the fraction step by step to finally get \(e^{2x}\).

Concluding the proof involves verifying that each step follows logically from the previous one, making sure that the transformation is both correct and necessary. This style of mathematical reasoning is essential for problem-solving in engineering mathematics.