Question

求这些数学导数题解题过程和答案

Answer 1

1、
(1) y=x^3+3^x-ln3
y'=3x^2+3^xln3
(2) y=arctan√x
y=1/[1+(√x)^2]×(√x)'
=1/(1+x)×1/(2√x)
=1/[2√x(1+x)]
(3) y=lnsinx
y'=1/sinx×(sinx)'
=1/sinx×cosx
=cotx
(4) y=x^sinx (x>0)
lny=sinxlnx
1/y×y'=cosxlnx+sinx×1/x
y'=y(cosxlnx+1/xsinx)
=x^sinx(cosxlnx+1/xsinx)
2、y=lnsin^2(1/x)
dy=1/[sin^2(1/x)]×[sin^2(1/x)]'dx
=1/[sin^2(1/x)]×2sin(1/x)×[sin(1/x)]'dx
=2/[sin(1/x)]×cos(1/x)×(1/x)'dx
=2cot(1/x)×(-1/x^2)dx
=-2cot(1/x)/x^2dx
3、ye^x+lny=1
y'e^x+ye^x+1/yy'=0
(e^x+1/y)y'=-ye^x
y'=-ye^x/(e^x+1/y)
=-y^2e^x/(ye^x+1)
即：dy/dx=-y^2e^x/(ye^x+1)