Problem 15 - Essential Calculus

Question

In Exercises \(15-24,\) find the derivative of the function. \(y=\operatorname{sech}(x+1)\)

Step-by-Step Solution

Answer

The derivative of the function \( y=\operatorname{sech}(x+1) \) is \( y'=- \operatorname{sech}(x+1) \operatorname{tanh}(x+1) \).

Step 1: Recall derivative of sech function

The derivative of the hyperbolic secant function, \( \operatorname{sech}(x) \), can be expressed as \( - \operatorname{sech}(x) \operatorname{tanh}(x) \) .

Step 2: Apply chain rule

According to the Chain Rule, for any functions \(f\) and \(g\), the derivative of the composite function \(f(g(x))\) is given by \(f'(g(x)) \cdot g'(x)\). So, here our \(f(x) = \operatorname{sech}(x) \) and \(g(x) = x+1\). The derivative of \(g(x)\) or \(g'(x)\) would be \(1\).

Step 3: Compute the derivative of the function

Now, we can compute the derivative as follows: put \(f'(g(x))\) as \(- \operatorname{sech}(x+1) \operatorname{tanh}(x+1)\) and \(g'(x)\) as \(1\). The composite function's derivative will be \(- \operatorname{sech}(x+1) \operatorname{tanh}(x+1) \cdot 1\), which simplifies to \(- \operatorname{sech}(x+1) \operatorname{tanh}(x+1)\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but are based on hyperbolas instead of circles. These functions include hyperbolic sine (\(\sinh(x)\)), hyperbolic cosine (\(\cosh(x)\)), hyperbolic tangent (\(\tanh(x)\)), and others like the hyperbolic secant (\(\operatorname{sech}(x)\)). They are often used in calculus and have properties and formulas parallel to those of trigonometric functions.

To understand hyperbolic functions better, consider their relation to exponential functions:

• The function \(\sinh(x)\) can be defined as \(\frac{e^x - e^{-x}}{2}\).
• Similarly, \(\cosh(x)\) can be given by \(\frac{e^x + e^{-x}}{2}\).
• The hyperbolic secant function is written as \(\operatorname{sech}(x) = \frac{1}{\cosh(x)}\).

These functions are different, yet they share some similar differentiation rules to their trigonometric counterparts. Knowing these functions and their derivatives is crucial in solving many calculus problems.

Chain Rule

The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. It provides a method to break down challenging differentiation tasks into simpler ones.

When you have a composite function like \(f(g(x))\), the Chain Rule states that its derivative is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function: \(f'(g(x)) \cdot g'(x)\).

Applying the Chain Rule

• Identify the outer and inner functions. For example, if \(y = \operatorname{sech}(x+1)\), then the outer function \(f(x) = \operatorname{sech}(x)\) and the inner function \(g(x) = x + 1\).
• Compute the derivatives of the inner and outer functions. The derivative \(f'(x)\) of the hyperbolic secant is \(-\operatorname{sech}(x)\operatorname{tanh}(x)\), and the derivative of \(g(x) = x + 1\) is simply 1.
• Multiply these derivatives as per the Chain Rule: \(f'(g(x)) \cdot g'(x) = -\operatorname{sech}(x+1)\operatorname{tanh}(x+1) \cdot 1\).

Mastering the Chain Rule is critical for success in calculus, especially when dealing with more complex functions.

Composite Functions

Composite functions combine two or more functions into a single function. This concept often appears in calculus, involving differentials and integrals. A function \(h(x)\) is a composite of \(f\) and \(g\) if \(h(x) = f(g(x))\).

Understanding composite functions is essential to apply the Chain Rule effectively. Whenever you encounter a composite function:

• Decompose it into identifiable outer and inner functions. In our exercise, \(\operatorname{sech}(x+1)\) can be seen as \(\operatorname{sech}(g(x))\) where \(g(x) = x + 1\).
• This decomposition helps in applying differentiation rules like the Chain Rule efficiently.
• Remember that each function in the composite can affect the outcome differently, so careful consideration is necessary when working with them.

Composite functions allow us to model more complex behaviors in mathematics, physics, engineering, and many other fields. Recognizing and working with them allows for deeper insights and solutions to intricate problems.