Problem 2 - Calculus

Question

From the definitions of \(\cosh x\) and \(\sinh x\), find their derivatives.

Step-by-Step Solution

Answer

The derivative of \(\cosh x\) is \(\sinh x\), and the derivative of \(\sinh x\) is \(\cosh x\).

Step 1: Understanding \\cosh x and \\sinh x

First, let's recall the definitions of the hyperbolic functions. The hyperbolic cosine function is defined as \[\cosh x = \frac{e^x + e^{-x}}{2}\]and the hyperbolic sine function is defined as \[\sinh x = \frac{e^x - e^{-x}}{2}.\]

Step 2: Derivative of \\cosh x

To find the derivative of \(\cosh x\), use the definition: \[\frac{d}{dx}\left(\cosh x\right) = \frac{d}{dx}\left(\frac{e^x + e^{-x}}{2}\right).\]Apply the differentiation rules:\[\frac{d}{dx}\left(\cosh x\right) = \frac{1}{2} \left( e^x - e^{-x} \right) = \sinh x.\]

Step 3: Derivative of \\sinh x

Now find the derivative of \(\sinh x\) using its definition:\[\frac{d}{dx}\left(\sinh x\right) = \frac{d}{dx}\left(\frac{e^x - e^{-x}}{2}\right).\]Apply the differentiation rules:\[\frac{d}{dx}\left(\sinh x\right) = \frac{1}{2} \left( e^x + e^{-x} \right) = \cosh x.\]

Step 4: Summarizing Results

The derivative of \(\cosh x\) is \(\sinh x\) and the derivative of \(\sinh x\) is \(\cosh x\). These results show the close relationship between the hyperbolic sine and cosine functions, analogous to trigonometric identities in calculus.

Key Concepts

Derivative

The derivative is a fundamental concept in calculus, representing the rate of change or slope of a function. When finding the derivative of a function, we are essentially determining how the function changes with respect to a variable.

• The derivative provides instantaneous rates of change, essential in physics, economics, and other fields.
• In the context of hyperbolic functions, we use standard differentiation rules such as the sum and chain rules. These methods help evaluate the derivatives of more complex expressions.
• The process of differentiating involves breaking down the function into simpler parts, finding the derivative of each, and then reassembling.

Computing derivatives for hyperbolic functions, such as \(\cosh x\) and \(\sinh x\), highlights their unique properties and how they relate to exponential functions.

sinh x

The hyperbolic sine function, denoted as \(\sinh x\), plays a prominent role in hyperbolic geometry, similar to its triangular namesake, the sine function in circular trigonometry. The formula for hyperbolic sine is given by:\[\sinh x = \frac{e^x - e^{-x}}{2}\]Some notable attributes include:

• Extends the exponential function by combining \(e^x\) and \(e^{-x}\), offering symmetrical properties around the origin.
• Connections to the hyperbolic curve, describing shapes and forces in physics and engineering.
• The derivative of \(\sinh x\) itself is an elegant expression that yields \(\cosh x\), showing the interrelation between the two core hyperbolic functions.

cosh x

Hyperbolic cosine, represented as \(\cosh x\), is akin to the circular cosine function but defined for hyperbolic angles. Its formal expression is:\[\cosh x = \frac{e^x + e^{-x}}{2}\]Important properties of \(\cosh x\) include:

• Symmetrical around the y-axis, this function forms a symmetrical curve reminiscent of celestial orbits.
• Forms the basis for hyperbolic graphs and helps model real-world phenomena like suspension bridges.
• Its derivative, \(\sinh x\), reinforces the intrinsic link between hyperbolic cosine and sine, reflecting their seamless interchangeability in differentiation.

By understanding \(\cosh x\) and its derivative, students can better grasp real-world applications and mathematical intricacies associated with hyperbolic functions.