Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\). $$ y=\cosh ^{2} x $$

Step-by-Step Solution

Answer

The derivative \(D_{x} y\) is \(2 \cosh(x) \sinh(x)\).

Step 1: Identify the derivative rule to use

The function given is a composition of functions, specifically \( \cosh(x)^2 \) , which is the square of the hyperbolic cosine function. To differentiate this, we will use the chain rule, which states that \( D_{x}[f(g(x))] = f'(g(x)) \cdot g'(x) \). In our case, \( f(u) = u^2 \) and \( g(x) = \cosh(x) \).

Step 2: Differentiate the outer function

Differentiate the outer function \( f(u) = u^2 \) with respect to \( u \). The derivative is \( f'(u) = 2u \). So, when we apply this to \( g(x) \), it becomes \( 2 \cosh(x) \).

Step 3: Differentiate the inner function

Next, differentiate the inner function \( g(x) = \cosh(x) \). The derivative of \( \cosh(x) \) with respect to \( x \) is \( \sinh(x) \), which gives us \( g'(x) = \sinh(x) \).

Step 4: Apply the chain rule

Now, apply the chain rule: \( f'(g(x)) \cdot g'(x) = 2 \cosh(x) \cdot \sinh(x) \). This results in the derivative \( D_{x} y = 2 \cosh(x) \sinh(x) \).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used for finding the derivative of composite functions. When a function is composed of two or more functions, you need to use the chain rule to differentiate it correctly.
The chain rule can be expressed as follows:

• If you have a composite function, say \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \).

For example, if you have a function \( y = \,\cosh^2(x) \), you can see it as \( f(u) = u^2 \) where \( u = \cosh(x) \). This means \( f'(u) = 2u \) and \( g(x) = \cosh(x) \), where \( g'(x) = \sinh(x) \). You then apply the chain rule to get the full derivative.
The chain rule is powerful because it allows you to break down complicated problems into simpler parts, making differentiation straightforward even for complex functions.

Hyperbolic Functions

Hyperbolic functions are analogs of the classical trigonometric functions but for the hyperbola, rather than the circle. They serve useful purposes in calculus, especially in dealing with integrals and derivatives.
The basic hyperbolic functions are:

• Hyperbolic cosine: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)
• Hyperbolic sine: \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)

These functions have derivatives quite similar to regular trigonometric functions:

• The derivative of \( \cosh(x) \) is \( \sinh(x) \).
• The derivative of \( \sinh(x) \) is \( \cosh(x) \).

In the context of the original problem, the function \( \cosh^2(x) \) requires differentiating using these properties to find \( \sinh(x) \), facilitating the use of the chain rule.

Differentiation Techniques

Differentiation refers to finding the derivative, or the rate of change, of a function. Several techniques exist for differentiation, each suited for different types of functions or variable dependencies. Some of the prominent techniques include:

• Basic differentiation: The technique used when dealing with standard power, polynomial or constant functions. For instance, \( \frac{d}{dx}[x^n] = nx^{n-1} \).
• Product rule: Useful when differentiating the product of two functions, \( (uv)' = u'v + uv' \).
• Chain rule: As discussed, used for composite functions like \( f(g(x)) \).
• Implicit differentiation: Comes in handy when direct differentiation with respect to a variable is complex.

In the given exercise, the focus is on using the chain rule for differentiating a composition of the hyperbolic cosine function. The specific differentiation technique applied here combines both understanding of hyperbolic function derivatives and the chain rule, resulting in an accurate result for the rate of change of \( y = \cosh^2(x) \).