如何用代数方法解答下面各题

Question

如题所述

Answer 1

（1）y=arcsin[tan(x^2)]
dy/dx={1/√{1-[tan(x^2)]^2}}*[tan(x^2)]'
={1/√{1-[tan(x^2)]^2}}*[sec(x^2)]^2*(x^2)'
={1/√{1-[tan(x^2)]^2}}*[sec(x^2)]^2*(2x)
={2x[sec(x^2)]^2}/√{1-[tan(x^2)]^2}
（2）(x^2)*e^(-3y)+y^3=6
2x*e^(-3y)+(x^2)*e^(-3y)*(-3dy/dx)+(3y^2)*dy/dx=0
2x-(3x^2)*dy/dx+(3y^2)*e^(3y)*dy/dx=0
[(3y^2)*e^(3y)-3x^2]*dy/dx=-2x
dy/dx=(2x)/[3x^2-(3y^2)*e^(3y)]
（3）y=(2-x)^(1/x)
lny=(1/x)*ln(2-x)
(1/y)*dy/dx=(-1/x^2)*ln(2-x)+(1/x)*[1/(x-2)]
dy/dx=y*[1/(x^2-2x)-ln(2-x)/(x^2)]
=[(2-x)^(1/x)]*[1/(x^2-2x)-ln(2-x)/(x^2)]
（4）y=∫(0,√x) ln(t^4+1)dt
dy/dx=(√x)'*ln[(√x)^4+1]
=(1/2√x)*ln(x^2+1)
=ln(x^2+1)/(2√x)