Question

求下列函数的导数求解

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Answer 1

(1)
y= [sec(e^(2x))]^3
y'
=3[sec(e^(2x))]^2 .d/dx(sec(e^(2x)))
=3[sec(e^(2x))]^2 .(sec(e^(2x))). tan(sec(e^(2x))) .d/dx(e^(2x))
=3[sec(e^(2x))]^2 .(sec(e^(2x))). tan(sec(e^(2x))) .(2e^(2x))
=6[sec(e^(2x))]^3 . tan(sec(e^(2x))) .e^(2x)
(2)
y= arctan[(1-x)/(1+x)]
y'
=[ 1/{1+[(1-x)/(1+x)]^2 } ] . d/dx [(1-x)/(1+x)]
=[ 1/{1+[(1-x)/(1+x)]^2 } ] . ( -2/(1+x)^2 )
= -2/[ (1+x)^2 +(1-x)^2 ]
=-1/(1+x^2)
(3)
y= ln [(x-2)^(1/3) / (x^2+1)^(1/2) ]
=(1/3)ln(x-2) - (1/2)ln(x^2+1)
y'
=(1/3)[1/(x-2)] - x/(x^2+1)
(4)
y= tan[ln(1+x^2)]
y'
=(sec[ln(1+x^2)])^2 .d/dx (ln(1+x^2))
=(sec[ln(1+x^2)])^2 . [ 2x/(1+x^2) ]
(5)
y= e^(x/lnx)
y'
=e^(x/lnx) . d/dx (x/lnx)
=e^(x/lnx) . [ (lnx- 1)/(lnx)^2 ]
(6)
y= (x/2)√(a^2-x^2) + (a^2/2)arcsin(x/a)
y'
=(1/2){ -x^2/√(a^2-x^2) + √(a^2-x^2) } +(a^2/2) [(1/a)/√(1- (x/a)^2)]
=(1/2)[ a^2/√(a^2-x^2) ] + (1/2) [a^2/√(x^2- a)^2]
= a^2/√(a^2-x^2)