( u v)(n)=∑k=0 n C n k u k vn−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∑n​Cnk​ukvn−k=Cn0​u(0)v(n)+Cn1​u(1)v(n−1)++Cnn​u(n)v(0)

绝对值求导视而不见(xa)′=axa − 1 (a为常数)(ax)′=axlna(ex)′=ex (logax)′=1x l n a (a>0,a≠1)(lnx)′=1x(ln∣x∣)′=1x (sinx)′=cosx(cosx)′=−sinx (arcsinx)′=1 1 −x2(arccosx)′=−1 1 −x2 (tanx)′=sec2x(cotx)′=−csc2x (arctanx)′=11 +x 2(arccotx)′=−11 +x 2 (secx)′=secxtanx(cscx)′=−cscxtanx (ln∣secx+tanx∣)′=secx(ln∣cscx−cotx∣′=cscx (ln(1+ x 2+ 1 ))′=1x2 + 1(ln(1+ x 2− 1 ))′=1x2 − 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(loga​x)′=xlna1​(a>0,a=1)(lnx)′=x1​(ln∣x∣)′=x1​(sinx)′=cosx(cosx)′=−sinx(arcsinx)′=1−x2 ​1​(arccosx)′=−1−x2 ​1​(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+1 ​1​(ln(1+x2−1 ​))′=x2−1 ​1​​

(a x )(n)=a x( l n a) n(e x )(n)=e x( s i n k x)(n)=k ns i n ( k x +π 2n ) ( c o s k x)(n)=k nc o s ( k x +π 2n ) ( l n x)(n)= ( − 1)(n−1)(n−1)!xn ( 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(1x+a )(n)= (−1)nn!(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+2π​n)(coskx)(n)=kncos(kx+2π​n)(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!​​

[ ( x − x0)n]( n ) =n![(x-\color{red}{x_0}\color{black})^n]^{(n)}=n=n!