Question

几个高数题1.y=x^(e^x)+e^(e^x),求dy2.x=e^(2t)*(cost)^2,y=e^(2t)*(si

Answer 1

1.dy=d(x^(e^x))+d(e^(e^x))

=x^(e^x)d(e^xlnx)+e^(e^x)d(e^x)

=x^(e^x)(lnx+1/x)e^xdx+e^(e^x)e^xdx

=[x^(e^x)(lnx+1/x)+e^(e^x)]e^xdx

2.∵dx=e^(2t)d(cos²t)+cos²td(e^(2t))

=-e^(2t)sin(2t)dt+2e^(2t)cos²tdt

=[cos²t-sin(2t)]e^(2t)dt

dy=e^(2t)d(sin²t)+sin²td(e^(2t))

=e^(2t)sin(2t)dt+2e^(2t)sin²tdt

=[sin²t+sin(2t)]e^(2t)dt

∴dy/dx=[sin²t+sin(2t)]/[cos²t-sin(2t)]

3.y'=[(x²-x+1)^x ]'

=(x²-x+1)^x*[xln(x²-x+1)]'

=(x²-x+1)^x*[ln(x²-x+1)+x(2x-1)/(x²-x+1)]