Question

Use the Taylor series for \(\sinh x\) and \(\cosh x\) to verify that \(\frac{d}{d x} \sinh x=\cosh x\)

Short answer

Question: Verify that the derivative of \(\sinh x\) with respect to x is equal to \(\cosh x\) using their Taylor series. Answer: After differentiating the Taylor series for \(\sinh x\) with respect to x, we obtained the Taylor series for \(\cosh x\), verifying that \(\frac{d}{dx}\sinh x = \cosh x\).

Step-by-step solution

Write the Taylor series for \(\sinh x\) and \(\cosh x\)

Recall that the Taylor series of a function \(f(x)\) around the point \(a\) is given by: $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n,$$ where \(f^{(n)}(a)\) denotes the \(n\)-th derivative of the function at the point \(a\). The Taylor series of \(\sinh x\) and \(\cosh x\) around \(0\) are: $$\sinh x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1},$$ $$\cosh x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}.$$

Differentiate the Taylor series for \(\sinh x\) with respect to \(x\)

To differentiate the Taylor series of \(\sinh x\), we need to apply the rules of differentiation term by term: $$\frac{d}{dx}\sinh x = \frac{d}{dx}\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}\frac{d}{dx}x^{2n+1}.$$ Using the power rule of differentiation, we get: $$\frac{d}{dx}x^{2n+1} = (2n+1)x^{2n}.$$ Thus, the derivative of the Taylor series for \(\sinh x\) with respect to \(x\) is: $$\frac{d}{dx}\sinh x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}.$$

Compare the derivative of the Taylor series for \(\sinh x\) with the Taylor series for \(\cosh x\)

Now, we've found that the derivative of the Taylor series for \(\sinh x\) is: $$\frac{d}{dx}\sinh x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}.$$ Comparing this with the Taylor series for \(\cosh x\), which is: $$\cosh x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n},$$ we can see that both series are identical. Therefore, we have verified that: $$\frac{d}{dx}\sinh x = \cosh x.$$

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogues to the traditional trigonometric functions, but unlike their circular counterparts, they are defined through the relationships of a hyperbola rather than a circle. The two most fundamental hyperbolic functions are the hyperbolic sine and cosine, denoted as \(\sinh x\) and \(\cosh x\), respectively.

They can be defined using exponential functions as follows: \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\). Interestingly, hyperbolic functions share many properties with their trigonometric counterparts, such as addition formulas and derivatives that are related to each other. These functions frequently appear in both theoretical and applied mathematics, particularly in relation to calculus and differential equations.

Series Expansion

A series expansion allows us to express functions as a sum of infinitely many terms. These terms are usually derived from the function's derivatives at a single point, giving us insight into the function's behavior around that point. The most familiar type of series expansion might be the Taylor series, which can represent an infinite variety of functions as long as they are sufficiently smooth (meaning they can be differentiated repeatedly).

The power of the Taylor series lies in its ability to approximate functions with polynomials. The more terms we include in the series, the closer our approximation will be to the actual function. For example, \(e^x\) and trigonometric functions can all be expressed in terms of their Taylor series expansions.

Derivative of Hyperbolic Sine

When it comes to derivatives of hyperbolic functions, a remarkable pattern imitates the relationship between sine and cosine in trigonometry. The derivative of the hyperbolic sine function \(\sinh x\) is intriguingly simple: it is the hyperbolic cosine function \(\cosh x\).

Mathematically, this relationship can be expressed as \(\frac{d}{dx}\sinh x = \cosh x\). Understanding the derivative of \(\sinh x\) is crucial in solving differential equations and in various applications involving hyperbolic functions, such as in the description of hanging cables or suspension bridges, known in physics as the catenary problem.

Power Rule of Differentiation

The power rule of differentiation is a fundamental technique used in calculus to derive the rate of change of functions involving powers of the variable. For a function \(f(x) = x^n\), where \(n\) is any real number, the power rule states that its derivative with respect to \(x\) is \(f'(x) = nx^{n-1}\).

This rule greatly simplifies the differentiation process, especially when dealing with polynomials or Taylor series. For example, in the Taylor series of hyperbolic functions, applying the power rule term by term allows us to differentiate the entire series and find the corresponding series of the derivative function.