In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(sinh z)

Verified step by step guidance

1. Identify the function given: \(y = \ln(\sinh z)\), where \(\sinh z\) is the hyperbolic sine function.
2. Recall the chain rule for differentiation: if \(y = \ln(u)\), then \(\frac{dy}{dz} = \frac{1}{u} \cdot \frac{du}{dz}\).
3. Set \(u = \sinh z\). Then find the derivative of \(u\) with respect to \(z\): \(\frac{du}{dz} = \cosh z\), since the derivative of \(\sinh z\) is \(\cosh z\).
4. Apply the chain rule: \(\frac{dy}{dz} = \frac{1}{\sinh z} \cdot \cosh z\).
5. Express the derivative as \(\frac{dy}{dz} = \frac{\cosh z}{\sinh z}\), which can also be written as \(\coth z\) if desired.

Key Concepts

Derivative of the Natural Logarithm Function

The derivative of ln(u), where u is a differentiable function of a variable, is given by (1/u) * (du/dx). This rule allows us to differentiate logarithmic functions by first identifying the inner function and then applying the chain rule.

Hyperbolic Sine Function and Its Derivative

The hyperbolic sine function, sinh(z), is defined as (e^z - e^(-z))/2. Its derivative with respect to z is cosh(z), which is (e^z + e^(-z))/2. Knowing this derivative is essential when differentiating expressions involving sinh.

Chain Rule in Differentiation

The chain rule is used to differentiate composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). This rule is crucial when differentiating ln(sinh z), as ln and sinh are nested functions.