设置分母有原则，简单因式才下放
n阶求导
( u v ) ( n ) = ∑ k = 0 n C n k u k v n − k = C n 0 u ( 0 ) v ( n ) + C n 1 u ( 1 ) v ( n − 1 ) + + C n n u ( n ) v ( 0 ) (uv)^{(n)}=\sum_{k=0}^nC_n^ku^{k}v^{n-k}\\ =C_n^0u^{(0)}v^{(n)}+C_n^1u^{(1)}v^{(n-1)}++C_n^nu^{(n)}v^{(0)} (uv)(n)=k=0∑nCnkukvn−k=Cn0u(0)v(n)+Cn1u(1)v(n−1)++Cnnu(n)v(0)
求导公式
绝对值求导视而不见 ( x a ) ′ = a x a − 1 ( a为常数 ) ( a x ) ′ = a x l n a ( e x ) ′ = e x ( l o g a x ) ′ = 1 x l n a ( a > 0 , a ≠ 1 ) ( l n x ) ′ = 1 x ( l n ∣ x ∣ ) ′ = 1 x ( s i n x ) ′ = c o s x ( c o s x ) ′ = − s i n x ( a r c s i n x ) ′ = 1 1 − x 2 ( a r c c o s x ) ′ = − 1 1 − x 2 ( t a n x ) ′ = s e c 2 x ( c o t x ) ′ = − c s c 2 x ( a r c t a n x ) ′ = 1 1 + x 2 ( a r c c o t x ) ′ = − 1 1 + x 2 ( s e c x ) ′ = s e c x t a n x ( c s c x ) ′ = − c s c x t a n x ( l n ∣ s e c x + t a n x ∣ ) ′ = s e c x ( l n ∣ c s c x − c o t x ∣ ′ = c s c x ( l n ( 1 + x 2 + 1 ) ) ′ = 1 x 2 + 1 ( l n ( 1 + x 2 − 1 ) ) ′ = 1 x 2 − 1 \color{red}{\text{绝对值求导视而不见}} \begin{array}{l} (x^a)'=ax^{a-1} (\text{a为常数}) \quad \quad (a^x)'=a^xlna \quad \quad (e^x)'=e^x \\ \\ (log_ax)'=\frac{1}{xlna} (a> 0,a\neq1) \quad \quad (lnx)'=\frac{1}{x} \quad \quad (ln|x|)'=\frac{1}{x} \\ \\ (sinx)'=cosx \quad (cosx)'=-sinx \\ \\ (arcsinx)'=\frac{1}{\sqrt{1-x^2}} \quad \quad (arccosx)'=-\frac{1}{\sqrt{1-x^2}} \\ \\ (tanx)'=sec^2{x} \quad \quad (cotx)'=-csc^2{x} \\ \\ (arctanx)'=\frac{1}{1+x^2} \quad \quad (arccotx)'=-\frac{1}{1+x^2} \\ \\ (secx)'=secxtanx \quad \quad (cscx)'=-cscxtanx \\ \\ (ln|secx+tanx|)'=secx \quad \quad (ln|cscx-cotx|'=cscx \\ \\ (ln(1+\sqrt{x^2+1}))'=\frac{1}{\sqrt{x^2+1}} \quad \quad (ln(1+\sqrt{x^2-1}))'=\frac{1}{\sqrt{x^2-1}} \\ \end{array} 绝对值求导视而不见(xa)′=axa−1(a为常数)(ax)′=axlna(ex)′=ex(logax)′=xlna1(a>0,a=1)(lnx)′=x1(ln∣x∣)′=x1(sinx)′=cosx(cosx)′=−sinx(arcsinx)′=1−x21(arccosx)′=−1−x21(tanx)′=sec2x(cotx)′=−csc2x(arctanx)′=1+x21(arccotx)′=−1+x21(secx)′=secxtanx(cscx)′=−cscxtanx(ln∣secx+tanx∣)′=secx(ln∣cscx−cotx∣′=cscx(ln(1+x2+1))′=x2+11(ln(1+x2−1))′=x2−11
n阶导数求导
( a x ) ( n ) = a x ( l n a ) n ( e x ) ( n ) = e x ( s i n k x ) ( n ) = k n s i n ( k x + π 2 n ) ( c o s k x ) ( n ) = k n c o s ( k x + π 2 n ) ( l n x ) ( n ) = ( − 1 ) ( n − 1 ) ( n − 1 ) ! x n ( x > 0 ) ( l n ( 1 + x ) ) ( n ) = ( − 1 ) n − 1 ( n − 1 ) ! ( 1 + x ) n ( x > − 1 ) [ ( x + x 0 ) m ] ( n ) = m ( m − 1 ) ( m − 2 ) … ( m − n + 1 ) ( x + x 0 ) m − n ( 1 x + a ) ( n ) = ( − 1 ) n n ! ( x + a ) n + 1 \begin{array}{l} \\ (a^x)^{(n)} = a^x(lna)^n \quad\quad (e^x)^{(n)}=e^x\\ \\ (sinkx)^{(n)} = k^nsin(kx+\frac{\pi}{2}n)\\ \\ (coskx)^{(n)} = k^ncos(kx+\frac{\pi}{2}n)\\ \\ (lnx)^{(n)} = (-1)^{(n-1)}\frac{(n-1)!}{x^n}\quad (x>0)\\ \\ (ln(1+x))^{(n)} = (-1)^{n-1}\frac{(n-1)!}{(1+x)^n}\quad(x>-1)\\ \\ [(x+x_0)^m]^{(n)}=m(m-1)(m-2)\dots(m-n+1)(x+x_0)^{m-n}\\ \\ (\frac{1}{x+a})^{(n)}=\frac{(-1)^nn!}{(x+a)^{n+1}}\\ \end{array} (ax)(n)=ax(lna)n(ex)(n)=ex(sinkx)(n)=knsin(kx+2πn)(coskx)(n)=kncos(kx+2πn)(lnx)(n)=(−1)(n−1)xn(n−1)!(x>0)(ln(1+x))(n)=(−1)n−1(1+x)n(n−1)!(x>−1)[(x+x0)m](n)=m(m−1)(m−2)…(m−n+1)(x+x0)m−n(x+a1)(n)=(x+a)n+1(−1)nn!
例
[ ( x − x 0 ) n ] ( n ) = n ! [(x-\color{red}{x_0}\color{black})^n]^{(n)}=n! [(x−x0)n](n)=n!