91Ó°ÊÓ

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola, as opposed to a circle. Among them, the cosh function, or hyperbolic cosine function, is significant. It is defined as

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

Just like the functions sine and cosine in trigonometry, hyperbolic functions have their own inverse functions. The inverse hyperbolic cosine, written as \( \cosh^{-1}(x) \), returns the value whose hyperbolic cosine is \( x \). It is important to note the range of \( \cosh^{-1} \), which is

• [0, \(\infty\)], for which it always yields non-negative values.

This property leads the hyperbolic cosine's inverse, combined with the identity \( \cosh^{-1}(\cosh(x)) = |x| \), to simplify easily when dealing with real numbers. Understanding this property helps in approaching problems involving the differentiation of hyperbolic functions.

The absolute value function is a fundamental mathematical concept represented as \( |x| \). It returns the non-negative value of \( x \) regardless of whether \( x \) is positive or negative. In simpler terms, it measures how far \( x \) is from zero on a number line.

There is a straightforward definition for the absolute value:

• If \( x \geq 0 \), then \(|x| = x \).
• If \( x < 0 \), then \(|x| = -x \).

This split in definition can easily be captured using the piecewise approach.

In terms of differentiation, the absolute value function poses a unique challenge but is manageable once you utilize its piecewise nature. For \( y = |x| \):

• \( \frac{dy}{dx} = 1 \) when \( x \geq 0 \), because the graph is increasing directly.
• \( \frac{dy}{dx} = -1 \) when \( x < 0 \), since the graph decreases as it approaches zero.

Through these observations, you can see how the absolute value function helps reveal details about the nature of a function under differentiation.

A piecewise function is made up of several sub-functions, each sub-function applies to a certain interval of the main function's domain. This is necessary when a function behaves differently based on the input value of \( x \).

In the exercise provided, the piecewise nature arises due to the presence of the absolute value function. As \( x \) shifts from being negative to non-negative, the function itself alters its form. This is where the definition splits as:

• For \( x \geq 0 \), the function behaves as \( y = x \).
• For \( x < 0 \), the function becomes \( y = -x \).

When differentiating piecewise functions, one crucial step is to evaluate the derivative for every piece separately. Generally, you will find constant derivatives over their respective intervals, as shown in the step-by-step solution:

• \( \frac{dy}{dx} = 1 \) when \( x \geq 0 \).
• \( \frac{dy}{dx} = -1 \) when \( x < 0 \).

These result from each piece being a linear function within its defined interval. Ultimately, piecewise functions play a crucial role in simplifying complex functions into manageable parts. This makes them extremely powerful in calculus and broader mathematics.