91Ó°ÊÓ

Logarithmic differentiation is a handy technique for deriving derivatives, especially when dealing with complex expressions that include logarithms. In the expression given, we have a logarithmic function, specifically \( \ln \cosh v \). Differentiating this term involves understanding how to apply the derivative rules for logarithmic functions.

When taking the derivative of \( \ln u \) with respect to \( u \), we obtain \( 1/u \). However, since \( \cosh v \) isn't a plain \( v \), we need to utilize the chain rule to take its derivative.

Remember, the derivative of \( \cosh v \) is \( \sinh v \), which helps in simplifying the derivative of \( \ln \cosh v \). By using the logarithmic differentiation technique, we benefit from simplifying complex multiplicative or chained expressions.

Hyperbolic functions, such as \( \cosh \) and \( \tanh \), resemble trigonometric functions, yet they bring their unique properties. Familiarizing yourself with these functions is essential for calculus operations like differentiation.

Each hyperbolic function's derivative has a specific pattern:

• The derivative of \( \cosh v \) is \( \sinh v \).
• The derivative of \( \tanh v \) is the square of the hyperbolic secant, represented as \( \text{sech}^2 v \).

These basic rules form the foundation needed for differentiating more complex expressions involving hyperbolic functions.

Consequently, to differentiate \( \tanh^2 v \), we use these rules along with the chain rule to reach the derivative effectively.

The chain rule is a fundamental concept for differentiating composite functions, especially when an inside function and outside function are involved.

In the given exercise, we encounter a situation in which the chain rule comes into play multiple times. Take \( \ln \cosh v \) as an example, where you first differentiate \( \ln u \) by \( 1/u \), but then you must subsequently multiply by the derivative of \( u \), which is \( \cosh v \).

Remember: the chain rule follows when differentiating \( f(g(v)) \), effectively:\[ \frac{d}{dv}f(g(v)) = f'(g(v)) \cdot g'(v) \] This principle ensures we account for the "chain" of dependencies from the inner function to the outer function, allowing us to find the derivative with respect to the main variable accurately.

Calculating derivatives accurately requires the right combination of rules and techniques, which we've applied to solve this exercise effectively.

Our goal was to derive \( y = \ln \cosh v - \frac{1}{2} \tanh^2 v \) with respect to \( v \). By using the skills we've discussed, such as identifying key functions, applying the chain rule, and simplifying our results, we reached the final outcome.

Initially, each part of the expression was differentiated independently; first the logarithmic term, then \( \tanh^2 v \). To finalize the solution, these derivatives were thoughtfully combined.

The crucial result rendered from all our calculations and simplifications was that \( \frac{dy}{dv} = \tanh^3 v \), a clear outcome of our systematic derivative calculation process.