In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
21. y = ln(cosh v) - 1/2 tanh²v

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Identify the function to differentiate: \(y = \ln(\cosh v) - \frac{1}{2} \tanh^{2} v\).

Recall the derivative formulas needed: the derivative of \(\ln(u)\) is \(\frac{1}{u} \frac{du}{dv}\), and the derivative of \(\tanh v\) is \(\operatorname{sech}^2 v\).

Differentiate the first term: \(\frac{d}{dv} \ln(\cosh v) = \frac{1}{\cosh v} \cdot \sinh v\) because \(\frac{d}{dv} \cosh v = \sinh v\).

Differentiate the second term using the chain rule: \(\frac{d}{dv} \left( \frac{1}{2} \tanh^{2} v \right) = \frac{1}{2} \cdot 2 \tanh v \cdot \operatorname{sech}^2 v\).

Combine the derivatives from both terms carefully, remembering to subtract the derivative of the second term from the first to find \(\frac{dy}{dv}\).

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주요 개념

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Derivative of Hyperbolic Functions

Hyperbolic functions such as sinh, cosh, and tanh have derivatives similar to trigonometric functions but with distinct signs. For example, the derivative of cosh(v) is sinh(v), and the derivative of tanh(v) is sech²(v). Understanding these derivatives is essential for differentiating expressions involving hyperbolic functions.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This rule is crucial when differentiating functions like ln(cosh v) where cosh v is inside the logarithm.

Derivative of Logarithmic Functions

The derivative of the natural logarithm function ln(u) with respect to its variable is 1/u times the derivative of u. This rule allows us to differentiate expressions like ln(cosh v) by first identifying the inner function and then applying the chain rule to find the overall derivative.

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.21. y = ln(cosh v) - 1/2 tanh²v