Problem 18 - Thomas Calculus

Question

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln (\cosh z)$$

Step-by-Step Solution

Answer

The derivative of \(y = \ln(\cosh z)\) is \(\tanh(z)\).

Step 1: Identify the Function Type

The function given is a composite function where the outer function is the natural logarithm, \(\ln(x)\), and the inner function is the hyperbolic cosine, \(\cosh(z)\). Our task is to differentiate \(y = \ln(\cosh z)\).

Step 2: Use Chain Rule

To differentiate a composite function, use the chain rule. The derivative of \(\ln(u)\) with respect to \(z\) is \(\frac{1}{u} \cdot \frac{du}{dz}\). Here, \(u = \cosh(z)\).

Step 3: Differentiate the Inner Function

Differentiate \(\cosh(z)\) with respect to \(z\). The derivative of \(\cosh(z)\) is \(\sinh(z)\).

Step 4: Differentiate the Outer Function

Apply the chain rule as derived in Step 2: the derivative of \(y = \ln(\cosh z)\) is given by \(\frac{1}{\cosh(z)} \cdot \sinh(z)\).

Step 5: Simplify the Expression

Simplify \(\frac{\sinh(z)}{\cosh(z)}\). This simplifies to \(\tanh(z)\) since \(\tanh(z) = \frac{\sinh(z)}{\cosh(z)}\). Thus, \(\frac{dy}{dz} = \tanh(z)\).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. Composite functions are functions within functions, and the chain rule helps us tackle these by breaking them down into simpler parts. If you have a function of the form \(y = f(g(x))\), where \(f\) and \(g\) are both differentiable functions, the chain rule states that the derivative \(\frac{dy}{dx}\) is given by:

• First, differentiate the outer function \(f\) with respect to its argument, \(g(x)\).
• Then, multiply by the derivative of the inner function \(g\) with respect to \(x\).

This can be expressed as: \[\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\]In our exercise, we apply the chain rule to \(y = \ln(\cosh z)\). Here, \(\ln(x)\) is the outer function and \(\cosh(z)\) is the inner function. By applying the chain rule, we find that the derivative is \(\frac{1}{\cosh(z)} \cdot \sinh(z)\). This highlights the elegance and power of the chain rule in breaking down complex differentiation into manageable steps.

Hyperbolic Functions

Hyperbolic functions, much like trigonometric functions, arise frequently in mathematics and physics but are related to hyperbolas rather than circles.

• Key hyperbolic functions include \(\sinh(z)\) and \(\cosh(z)\), defined by the equations \(\sinh(z) = \frac{e^z - e^{-z}}{2}\) and \(\cosh(z) = \frac{e^z + e^{-z}}{2}\).
• These functions have derivatives very similar to their circular counterparts:

When we differentiate \(\cosh(z)\), the result is \(\sinh(z)\) and vice versa, the derivative of \(\sinh(z)\) is \(\cosh(z)\). This symmetry simplifies calculations and is instrumental in solving complex differential equations.
In the problem \(y = \ln(\cosh z)\), knowing the derivative of \(\cosh(z)\) is \(\sinh(z)\) is crucial, as it allows us to apply the chain rule effectively. These hyperbolic functions are encountered in scenarios ranging from solving Laplace's equation in physics to describing the shape of a hanging cable, known as the catenary.

Derivative Simplification

Simplifying derivatives is an important step to make the functions cleaner and easier to interpret or use in further calculations. After applying the chain rule in our example, we arrived at the derivative \(\frac{dy}{dz} = \frac{\sinh(z)}{\cosh(z)}\).

• This expression is already a type of hyperbolic function called \(\tanh(z)\), which is short for the hyperbolic tangent.
• The equation \(\tanh(z) = \frac{\sinh(z)}{\cosh(z)}\) simplifies our result into a more recognizable and compact form.

By expressing the derivative in terms of hyperbolic tangent, it not only makes computations easier but also offers insight into the behavior of the derivative across different domains. Simplification often utilizes identities, like those of hyperbolic functions, to express derivatives succinctly, aiding in both theoretical understanding and practical application.