(uv)' = u'v+uv '，
(uv)'' = u''v+2u'v'+uv ' '
According to mathematical induction, ...
The meaning of each symbol
σ-summation symbol
C (n, k)- combination symbol, that is, the combination of n and k.
N-k-n-k derivative of u.
The k-th derivative of v (k)-v.
This formula is similar to binomial theorem in permutation and combination, and the number of powers in binomial theorem is changed to the number of derivatives.
(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.
Extended data:
The derivation process of Leibniz formula
If there are functions u=u(x) and v=v(x), and both of them have n-order derivatives at x point, then obviously,
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 ' ' '
Baidu Encyclopedia-Leibniz Formula