Leibniz formula, like binomial theorem, is used to find the higher derivative of f(x)*g(x).
(ultraviolet)' = u' v+ ultraviolet'.
(uv)'' = u''v+2u'v'+uv ' ' .
According to mathematical induction, ...
(uv) first derivative =u first derivative times v+u times v first derivative.
(uv) Second derivative =u second derivative times v+2 times U first derivative times V first derivative +u times V second derivative.
(uv) third derivative =u third derivative times v+3 times u second derivative times v first derivative +3 times u first derivative times v second derivative +u times v third derivative.
If there are functions u=u(x) and v=v(x), and both of them have n-order derivatives at point X, then it is obvious.
U (x) v (x) also has an nth derivative at x, (u v) (n) = u (n) v (n).
As for the n-order derivative of u(x) × v(x), it is more complicated. According to the basic deduction rules and formulas, we can get:
(ultraviolet)' = u'v+ultraviolet'.
(uv)'' = u''v + 2u'v' + uv ' ' .
(uv)''' = u'''v + 3u''v' + 3u'v'' + uv ' ' .