I have been studying calculus for past few months and through the time I have been using the so called prime notation.I have been studying from Spivaks Calculus for those of you who are familiar with the book.
My problem now is that I want to study some differential equations so I can study Lagrangian and Hamiltonian mechanics eventually,but the books that teach differential equations mostly use Leibniz notation which I am not familiar with.
Could some of you point me in direction where I can learn to use Leibniz notation and convert it to prime notation,and other way around as well?
Also I would appreciate if someone would layout examples of chain rule,multiplication rule and other basic differentiation rules expressed in Leibniz notation in comparision to prime notation.
Thanks in advance
First Derivative:
$f'(x) = \dfrac {d}{dx} f(x) = \dfrac {df}{dx}(x) = \dfrac {df(x)}{dx} = \left.\dfrac {df}{dx}\right|_x$ and often also written simply as $\dfrac {df}{dx}$
Second Derivative:
$f''(x) = \dfrac {d^2}{dx^2}f(x) = \dfrac {d^2f}{dx^2}(x) = \dfrac {d^2f(x)}{dx^2} = \left.\dfrac {d^2f}{dx^2}\right|_x$ and often also written simply as $\dfrac {d^2f}{dx^2}$
The extension to higher derivatives should be obvious.
Chain rule: Let $y = g(x)$.
$(f\circ g)'(x) = f'(g(x))g'(x) \iff \dfrac {df}{dx} = \dfrac {df}{dy}\dfrac {dy}{dx}$
Product rule:
$(fg)'(x) = f'(x)g(x) + f(x) g'(x) \iff \dfrac d{dx} \big(f(x)g(x)\big) = \dfrac {df(x)}{dx}g(x) + f(x)\dfrac {dg(x)}{dx}$