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 is \( \tanh(z) \).

Step 1: Identify the Function

The given function is \( y = \ln(\cosh(z)) \). We need to differentiate \( y \) with respect to \( z \).

Step 2: Apply the Chain Rule

To differentiate \( y \), we'll use the chain rule for the composition of functions. The outer function is \( \ln(u) \) where \( u = \cosh(z) \), and the inner function is \( \cosh(z) \).

Step 3: Differentiate the Outer Function

The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). So, the derivative of \( \ln(\cosh(z)) \) with respect to \( \cosh(z) \) is \( \frac{1}{\cosh(z)} \).

Step 4: Differentiate the Inner Function

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

Step 5: Apply the Chain Rule

Multiply the derivatives from Step 3 and Step 4 together: \( \frac{1}{\cosh(z)} \times \sinh(z) = \frac{\sinh(z)}{\cosh(z)} \).

Step 6: Simplify the Expression

\( \frac{\sinh(z)}{\cosh(z)} \) can be simplified to \( \tanh(z) \), since \( \tanh(z) = \frac{\sinh(z)}{\cosh(z)} \).

Step 7: Conclusion

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

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. It allows us to break down complex functions into simpler components, making differentiation more manageable. Essentially, the chain rule states that to differentiate a composite function, you differentiate the outer function with respect to the inner function and multiply by the derivative of the inner function with respect to the variable.
For example, if you have a function like \(y = \ln(\cosh(z))\), you identify the outer function as \(\ln(u)\) and the inner function as \(u = \cosh(z)\). You first differentiate the logarithm, yielding \(\frac{1}{\cosh(z)}\), and then multiply it by the derivative of the hyperbolic cosine, which is \(\sinh(z)\).
Together, these steps produce \(\frac{\sinh(z)}{\cosh(z)}\). This combination process is the heart of using the chain rule efficiently.

• It breaks down complex differentiation.
• Provides a step-by-step approach.
• Widely applicable in calculus problems.

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but for the hyperbola rather than the circle. They include \(\sinh(z)\), \(\cosh(z)\), and \(\tanh(z)\), among others.
These functions are essential in various fields like engineering, physics, and mathematics for describing hyperbolic geometry and systems.
The hyperbolic cosine \(\cosh(z)\) and the hyperbolic sine \(\sinh(z)\) are defined as follows:\[\cosh(z) = \frac{e^z + e^{-z}}{2}\] and \[\sinh(z) = \frac{e^z - e^{-z}}{2}\].
The relationship between \(\sinh\) and \(\cosh\) gives us \(\tanh(z) = \frac{\sinh(z)}{\cosh(z)}\), which is the simplified form of the derivative we calculated earlier.

• Hyperbolic functions arise naturally in many physical situations.
• They have interesting properties, such as the identity \(\cosh^2(z) - \sinh^2(z) = 1\).
• Their derivatives mirrors those of trigonometric functions: \(\frac{d}{dz}\cosh(z) = \sinh(z)\) and \(\frac{d}{dz}\sinh(z) = \cosh(z)\).

Logarithmic Differentiation

Logarithmic differentiation is a technique used when dealing with functions that are products, quotients, or powers involving variable exponents. It simplifies the differentiation process by taking advantage of the properties of logarithms.
In our problem, we differentiated \(y = \ln(\cosh(z))\). This function already involves a logarithm, but knowing how logarithmic differentiation works can simplify other complex expressions.
The main idea consists of taking the natural logarithm of both sides and then differentiating. This approach transforms multiplicative relationships into additive ones and exponents into coefficients, making differentiation straightforward.
While not directly used in our example, understanding logarithmic differentiation helps enhance comprehension of logarithms' flexible nature.

• Useful for differentiating complex multiplication or division.
• Makes handling exponentials easier.
• Simplifies complicated derivatives, especially with variable exponents or non-standard bases.