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The chain rule is a crucial concept in calculus used for differentiating composite functions. A composite function is when one function is applied to the result of another function. In simpler terms, if you have a function like \( h(x) = f(g(x)) \), you use the chain rule to find the derivative.
Here’s a step-by-step approach to using the chain rule:

• Identify the outer function, \( f(u) \), and the inner function, \( u(x) \). For our example, \( h(x) = \ln(\cosh x) \), the outer function is \( f(u) = \ln(u) \), and the inner function is \( u(x) = \cosh(x) \).
• Differentiate the outer function concerning the inner function, which gives us \( f'(u) = \frac{1}{u} \).
• Differentiate the inner function concerning \( x \), giving us \( u'(x) = \sinh(x) \).
• Finally, multiply these two derivatives: \( h'(x) = f'(u(x)) \cdot u'(x) = \frac{1}{\cosh(x)} \cdot \sinh(x) \).

The result is the derivative of the composite function! Understanding the chain rule is vital because it allows you to tackle more complex function differentiation by breaking them into manageable pieces.

Hyperbolic functions are a set of functions that are analogues of the well-known trigonometric functions but for a hyperbola, just as trigonometric functions are based on a circle. Two essential hyperbolic functions are \( \sinh(x) \) and \( \cosh(x) \).
Here's a quick outline of these key hyperbolic functions:

• For \( \cosh(x) \), it's defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).
• \( \sinh(x) \) is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).

These functions have derivatives similar to trigonometric functions which make them simple and predictable. For example, the derivative of \( \cosh(x) \) is \( \sinh(x) \), and vice versa, \( \frac{d}{dx}[\sinh(x)] = \cosh(x) \).
One critical identity to remember is the relationship \( \frac{\sinh(x)}{\cosh(x)} = \tanh(x) \), which simplifies calculations as exemplified in the problem by transforming \( \frac{\sinh(x)}{\cosh(x)} \) into \( \tanh(x) \). Understanding these identities and properties can help simplify complex calculus problems.

Logarithmic differentiation is a method used to differentiate functions in which the traditional differentiation approach might be cumbersome, especially in handling products or quotients of functions.
Here’s how logarithmic differentiation is generally approached:

• Start by taking the natural logarithm on both sides of the function you wish to differentiate, where applicable. This simplifies expressions that are products or powers.
• Differentiate the resulting logarithmic equation using standard differentiation rules, including the chain rule. When you differentiate \( \ln(f(x)) \), the derivative is \( \frac{1}{f(x)} \cdot f'(x) \).

In the exercise provided, \( h(x) = \ln(\cosh x) \), the logarithmic function applies directly without needing to simplify products or powers, allowing straightforward differentiation using standard rules. Using logarithmic differentiation is advantageous as it changes multiplicative relationships into additive ones, simplifying complex functions into easier forms for differentiation. This often results in cleaner, more approachable calculus.