Leibniz theorem for successive differentiation

Question

I want to study Leibniz's theorem of successive differentiation and unfortunately, I found no books online to study and have some rigorous practice on the topic. I have some local books but they are extremely vague and not stated with clarity. Can anyone recommend me a book where it contains the theory with proper clarity and in-depth exercises?

Answer 1

Chapter 9, Section 3 of the 3rd edition of Boas's Mathematical Methods in the Physical Sciences gives some explicit examples of the use of Leibniz's rule for differentiating products, and assigns a few problems on it.

Section 1.4 of the 7th edition of Arfken, Weber, and Harris's Mathematical Methods for Physicists has the reader prove as an exercise Leibniz's rule for the repeated differentiation of a product, and then uses it in a few places throughout the text. You might find some intuition there.

The reference on the relevant Wikipedia page is for Olver's Applications of Lie Groups to Differential Equations, but I'm not familiar with it.

Answer 2

Here is the main calculation: \begin{align*} (f\cdot g)'(x)&=\lim_{h\to 0}\dfrac{f(x+h)g(x+h)-f(x)g(x)}{h}\\ &=\lim_{h\to 0}\dfrac{1}{h}\left(f(x+h)(g(x+h)-g(x))+(f(x+h)-f(x))g(x)\right)\\ &=\lim_{h\to 0}f(x+h)\dfrac{g(x+h)-g(x)}{h}+\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}g(x)\\ &=\left(\lim_{h\to 0}f(x+h\right)\cdot \lim_{h\to 0}\dfrac{g(x+h)-g(x)}{h}+\left(\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}\right)g(x)\\ &=f(x)\cdot g'(x)+f'(x)\cdot g(x) \end{align*}