Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall the derivative of hyperbolic sine

The derivative of the hyperbolic sine function \(\sinh(x)\) is given by \(\cosh(x)\).

Step 2: Use the Chain Rule for derivatives

Since \(y = \sinh^2(x)\), rewrite it as \(y = (\sinh(x))^2\). To find \(D_{x} y\), use the chain rule: \(D_{x} y = 2\cdot\sinh(x)\cdot D_{x}(\sinh(x))\).

Step 3: Calculate the derivative using chain rule result

Using the result of Step 1, substitute \(D_{x}(\sinh(x)) = \cosh(x)\) into the chain rule expression: \[ D_{x} y = 2\cdot \sinh(x) \cdot \cosh(x) \].

Step 4: Simplify the expression

The expression \(2\cdot \sinh(x) \cdot \cosh(x)\) can be simplified using the identity for \(\sinh(2x) = 2\cdot \sinh(x) \cdot \cosh(x)\), resulting in \(D_{x} y = \sinh(2x)\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions play a crucial role in calculus differentiation, much like their trigonometric counterparts. These functions are defined using exponential functions and have distinctive properties.
For hyperbolic sine, or \(\sinh(x)\), the function is expressed as:

• \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)

This hyperbolic function shares similarities with the sine function in trigonometry but applies different mathematical properties.
Hyperbolic cosine, or \(\cosh(x)\), complements \(\sinh(x)\) and is expressed as:

• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)

One interesting property of hyperbolic functions is their symmetry and relationship to the exponential functions of complex numbers.
In calculus, knowing the derivatives of these functions helps solve problems involving hyperbolic expressions and provide easy pathways to simplifying complex equations.

Chain Rule

The chain rule is a fundamental principle in calculus differentiation, helping when dealing with composite functions. It provides a method to differentiate a function that is composed of other functions.
In essence, the chain rule can be stated simply as: if you have a composite function \(f(g(x))\), the derivative \(D_x f(g(x)) = f'(g(x)) \cdot g'(x)\).
This rule emphasizes that to differentiate a composite function, you must:

• Take the derivative of the outer function \(f\) at the inner function \(g(x)\).
• Then multiply by the derivative of the inner function \(g(x)\).

This principle was aptly applied in the solution to differentiate \(y = \sinh^2(x)\) by rewriting it as \(y = (\sinh(x))^2\).
By applying the chain rule, we found that the derivative with respect to \(x\) involves both the original function's form and its derivative, providing a structured approach to handle more complex derivatives.

Derivatives

Differentiation is a core activity in calculus, focusing on finding how a function changes at any point, or in other words, its rate of change.
When we differentiate a function, we look for its derivative which indicates the slope of the function at any given point.
Hyperbolic functions, such as \(\sinh(x)\), have unique derivatives that are essential for solving calculus problems.

• The derivative of \(\sinh(x)\) is \(\cosh(x)\).
• This makes \(\cosh(x)\) a crucial part of any differentiation involving \(\sinh(x)\).

The example problem asked for the derivative of \(y = \sinh^2(x)\), involving both recognizing it as \((\sinh(x))^2\) and applying derivative rules like the chain rule.
Successfully differentiating involves both understanding these rules and recognizing patterns in the function's structure, thus enabling problem-solving in calculus.