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Chapter 12: Problem 1

By Leibniz' rule, write the formula for \(\left(d^{n} / d x^{n}\right)(u v)\).

Short Answer

Expert verified
The nth derivative formula by Leibniz's rule is \[\frac{d^n}{dx^n}(u v) = \sum_{k=0}^{n} \binom{n}{k} \frac{d^{n-k}u}{dx^{n-k}} \frac{d^k v}{dx^k}.\]

Step by step solution

01

Understand Leibniz's Rule

Leibniz's rule is used to find the nth derivative of a product of two functions. It’s a generalization of the product rule for derivatives.
02

Recall the Formula

The formula for the nth derivative of the product of two functions is: \[\frac{d^n}{dx^n}(u v) = \sum_{k=0}^{n} \binom{n}{k} \frac{d^{n-k}u}{dx^{n-k}} \frac{d^k v}{dx^k}.\] This means you take the sum of the products of the binomial coefficient, the (n-k)th derivative of u, and the kth derivative of v for k ranging from 0 to n.
03

Explain the Components

In the formula, \(\binom{n}{k}\) is the binomial coefficient, which is calculated as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Additionally, \( \frac{d^{n-k}u}{dx^{n-k}} \) is the (n-k)th derivative of u with respect to x, and \( \frac{d^{k}v}{dx^{k}} \) is the kth derivative of v with respect to x.
04

Summarize the Application

To apply Leibniz's rule, compute the necessary derivatives of u and v, then multiply these derivatives by the corresponding binomial coefficients, and finally sum all these products from k = 0 to n.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

product rule
The product rule is a fundamental concept in calculus used to find the derivative of the product of two functions. For functions u(x) and v(x), the product rule states that: \[ \frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx} \]. This tells us that the derivative of a product is not simply the product of the derivatives. Instead, we multiply each function by the derivative of the other function and add the results.
Leibniz's rule generalizes this idea to the nth derivative of a product. Instead of using just the first derivatives, we use higher-order derivatives with binomial coefficients to find the nth derivative of the product. This advanced rule allows us to handle more complex scenarios where the functions and their derivatives need to be analyzed in detail.

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Most popular questions from this chapter

Show that the set of functions \(\sin n x\) is not a complete set on \((-\pi, \pi)\) by trying to expand the function \(f(x)=1\) on \((-\pi, \pi)\) in terms of them.

By power series, solve the Hermite differential equation $$y^{\prime \prime}-2 x y^{\prime}+2 p y=0$$ You should find an \(a_{0}\) series and an \(a_{1}\) series as for the Legendre equation in Section 2 Show that the \(a_{0}\) series terminates when \(p\) is an even integer, and the \(a_{1}\) series terminates when \(p\) is an odd integer. Thus for each integer \(n\), the differential equation (22.14) has one polynomial solution of degree \(n\). These polynomials with \(a_{0}\) or \(a_{1}\) chosen so that the highest order term is \((2 x)^{n}\) are the Hermite polynomials. Find \(H_{0}(x), H_{1}(x),\) and \(H_{2}(x) .\) Observe that you have solved an eigenvalue problem (see end of Section 2 ), namely to find values of \(p\) for which the given differential equation has polynomial solutions, and then to find the corresponding solutions (eigenfunctions).

(a) The generating function for Bessel functions of integral order \(p=n\) is $$ \Phi(x, h)=e^{(1 / 2) x\left(h-h^{-1}\right)}=\sum_{n=-\infty}^{\infty} h^{n} J_{n}(x) $$ By expanding the exponential in powers of \(x\left(h-h^{-1}\right)\) show that the \(n=0\) term is \(J_{0}(x)\) as claimed. (b) Show that $$ x^{2} \frac{\partial^{2} \Phi}{\partial x^{2}}+x \frac{\partial \Phi}{\partial x}+x^{2} \Phi-\left(h \frac{\partial}{\partial h}\right)^{2} \Phi=0 $$ Use this result and \(\Phi(x, h)=\sum_{n=-\infty}^{\infty} h^{n} J_{n}(x)\) to show that the functions \(J_{n}(x)\) satisfy Bessel's equation. By considering the terms in \(h^{n}\) in the expansion of \(e^{(1 / 2) x\left(h-h^{-1}\right)}\) in part (a), show that the coefficient of \(h^{n}\) is a series starting with the term \((1 / n !)(x / 2)^{n}\). (You have then proved that the functions called \(J_{n}(x)\) in the expansion of \(\Phi(x, h)\) are indeed the Bessel functions of integral order previously defined by (12.9) and (13.1) with \(p=n\).)

Express each of the following polynomials as linear combinations of Legendre polynomials. Hint: Start with the highest power of \(x\) and work down in finding the correct combination. \(x^{4}\)

Solve the following differential equations by series and also by an elementary method and verify that your solutions agree. Note that the goal of these problems is not to get the answer (that's easy by computer or by hand) but to become familiar with the method of series solutions which we will be using later. Check your results by computer. $$x y^{\prime}=y$$
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