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y lntAnx 2求导数过程

y= ln(tan(x/2))y' = [1/tan(x/2)].[tan(x/2)]'= [1/tan(x/2)].(sec(x/2))^2 .(x/2)'= (1/2)[cot(x/2)]^2

令y=lnu,u=tanv.v=x/2所以y'=(lnu)'(tanv)'(x/2)' =(1/u)(sinv/cosv)'(1/2) =(1/2)[1/(tanx/2)]{[(sinv)'cosv-sinv(cosv)']/(cosv)^2} =(1/2)[1/(tanx/2)]{[(cosv)^2+(sinv)^2]/(cosv)^2} =(1/2)[1/(tanx/2)][1/(cosx/2)^2] =(1/2)[(cosx/2)/(sinx/2)][1/(cosx/2)^2] =(1/2){1/[(sinx/2)(cosx/2)]} =1/[2(sinx/2)(cosx/2)] =1/sinx

y=lntanx/2 y'=1/tanx/2 *(tanx/2)' =1/tanx/2 *sec^2(x/2)*(x/2)' =sec^2(x/2)/(2tanx/2)

y = ln tanx dy / dx = d(ln tanx) / d(tanx) * d(tanx) / dx= 1 / tanx * secx= 2 csc(2 x) dy / dx = 2 * dcsc(2 x) / d(2 x) * d(2 x) / dx= 2 * -csc(2 x) cot(2 x) * 2= -4 csc(2 x) cot(2 x) 或 = secx - cscx

1/(tanx/2)*(1/(cosx/2)^2)*1/2=1/2sin(x/2)cos(x/2)=1/sinx

y=ln(tanx/2)y'=1/tan(x/2)*sec^2(x/2)*(1/2)=1/sinx

=1/tan(x/2)*[tan(x/2)]'=cos(x/2)/sin(x/2)*sec(x/2)*(x/2)'=cos(x/2)/sin(x/2)*1/cos(x/2)*1/2=1/[2sin(x/2)cos(x/2)]=1/sinx

y=lntanx 则y的导数y'=1/(tanx)*(tanx)'=1/tanx*(secx)^2=cosx/sinx*1/(cosx)^2=1/(sinxcosx)=2/sin2x

令t=tan(x/2) t'=1/2*sec^2(x/2) 在y'=ln(tan(x/2))' =(lnt)'*t' =1/t*1/2*sec^2(x/2) =cot(x/2)*1/2*sec^2(x/2) =1/[2cos(x/2)sin(x/2)]

y=lntan(x/2) y'=[1/tan(x/2)]*[tan(x/2)]'=ctg(x/2)*sec^2(x/2)*(x/2)'=(1/2)ctg(x/2)*sec^2(x/2).=1/[2sin(x/2)cos(x/2)]=1/sinx=cscx.

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