arctan+x+y+的导数
函数y=arctan(3x+1)+2x的一阶和二阶三阶导数计算
主要内容:
本文主要用复合函数、和函数和函数商求导法则,并用幂函数、反正切函数的导数公式,介绍函数y=arctan(3x+1)+2x的三阶导数计算步骤。
导数公式:
本题主要用到的导数公式如下,其中c为常数:
A.若函数y=c,则导数dy/dx=0;
B.若函数y=cx,则导数dy/dx=c;
C.若函数y=arctanx,则导数dy/dx=1/(1+x^2)。
一阶导数计算:
因为:y=arctan(3x+1)+2x,由反正切和一次函数导数公式有:
所以:dy/dx=3/[1+(3x+1)^2]+2。
二阶导数计算:
因为:dy/dx=3x /[1+(3x+1)^2]+2,由函数商的求导法则有:
所以:d^2y/dx^2=-3*2(3x+1)*3/[1+(3x+1)^2]^2+0,
=-18(3x+1)/ [1+(3x+1)^2]^2。
三阶导数计算:
因为: d^2y/dx^2=-18 (3x+1)/ [1+(3x+1)^2]^2,
所以:
d^2y/dx^2=-18*{3[1+(3x+1)^2]^2-(3x+1)*2*[1+(3x+1)^2]*6(3x+1)}/ [1+(3x+1)^2]^4
=-18*{3 [1+(3x+1)^2]-(3x+1)*2*6 (3x+1)}/ [1+(3x+1)^2]^3
=-18*3{ [1+(3x+1)^2]-4(3x+1)(3x+1)}/ [1+(3x+1)^2]^3
=-18*3{ [1+(3x+1)^2]-4(3x+1)^2}/ [1+(3x+1)^2]^3
\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
=18*3 [3(3x+1)^2-1] / [1+(3x+1)^2]^3。
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