函数的求导或积分 急急急!
1、y=√(2x)e^(-x) 求y的导数
y'=[√(2x)e^(-x)]'
=[√(2x)]'*e^(-x)+√(2x)*[e^(-x)]'
=√2*(1/2)*[1/√x]*e^(-x)+√2*√x*e^(-x)*(-1)
=√2*e^(-x)*[(1/2√x)-√x]
2、y=3^(x/lnx) 求y的导数
y'=ln3*3^(x/lnx)*(x/lnx)'
=ln3*3^(x/lnx)*[(lnx-1)/(lnx)^2]
3、y=arccot[(1-sinx)/(1+sinx)]求y的导数
y'=-1/{1+[(1-sinx)/(1+sinx)]^2}*[(1-sinx)/(1+...全部
1、y=√(2x)e^(-x) 求y的导数
y'=[√(2x)e^(-x)]'
=[√(2x)]'*e^(-x)+√(2x)*[e^(-x)]'
=√2*(1/2)*[1/√x]*e^(-x)+√2*√x*e^(-x)*(-1)
=√2*e^(-x)*[(1/2√x)-√x]
2、y=3^(x/lnx) 求y的导数
y'=ln3*3^(x/lnx)*(x/lnx)'
=ln3*3^(x/lnx)*[(lnx-1)/(lnx)^2]
3、y=arccot[(1-sinx)/(1+sinx)]求y的导数
y'=-1/{1+[(1-sinx)/(1+sinx)]^2}*[(1-sinx)/(1+sinx)]'
=-(1+sinx)^2/[(1+sinx)^2+(1-sinx)^2]*{[-cosx(1+sinx)-(1-sinx)*cosx]/(1+sinx)^2}
=-(1+sinx)^2/[2(1+sin^2 x)]*[-2cosx/(1+sinx)^2]
=cosx/(1+sin^2 x)
4、y=ln(cot(x/2))求y的二次导数
y'=1/[cot(x/2)]*[cot(x/2)]'
=tan(x/2)*[-csc^2(x/2)]*(1/2)
=-(1/2)*[sin(x/2)/cos(x/2)]*[1/sin^2(x/2)]
=-1/[2sin(x/2)cos(x/2)]
=-1/sinx
=-cscx
则,y''=(-cscx)'=cscx*cotx
5、f(x)=x/(√(1-x²)) 求f''(0)
f'(x)=[√(1-x^2)-x*(1/2)*1/√(1-x^2)*(-2x)]/(1-x^2)
=[√(1-x^2)+x^2/√(1-x^2)]/(1-x^2)
=[(1-x^2)+x^2]/(1-x^2)^(3/2)
=1/(1-x^2)^(3/2)
=(1-x^2)^(-3/2)
则,f''(x)=(-3/2)*(1-x^2)^(-5/2)*(-2x)=3x*(1-x^2)^(-5/2)
所以,f''(0)=0
6、x/y=arctany 求dy
x/y=arctany
===> yarctany=x
===> d(yarctany)=dx
===> dy*arctany+y*[1/(1+y^2)]dy=dx
===> [arctany+y/(1+y^2)]dy=dx
===> [(1+y^2)arctany+y]dy/(1+y^2)=dx
===> dy=(1+y^2)dx/[(1+y^2)arctany+y]
7、y=x^(2x) 求y'
y=x^(2x)
则,lny=2xlnx
===> (1/y)*y'=2(lnx+1)
===> y'=2y(lnx+1)
===> y'=2(lnx+1)*x^(2x)
8、y=(x√(1-x)/(三次√(1+x)) 求y'
y'={[x√(1-x)]'*(1+x)^(1/3)-x√(1-x)*(1/3)*(1+x)^(-2/3)}/[(1+x)^(2/3)]
={[√(1-x)+x*(1/2)*(1-x)^(-1/2)*(-1)]*(1+x)^(1/3)-(x/3)√(1-x)*(1+x)^(-2/3)}/[(1+x)^(2/3)]
={[√(1-x)-(x/2)*(1-x)^(-1/2)]*(1+x)^(1/3)-(x/3)*√(1-x)*(1+x)^(-2/3)}/[(1+x)^(2/3)]。
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