作业帮问一道参数方程阶导数的题目,求出以下参数方程所确定的函数的指定阶函数.已知:x=ln(1+t^2)y=t-arctan(t),求d^3y/dx^3
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问一道参数方程阶导数的题目,求出以下参数方程所确定的函数的指定阶函数.已知:x=ln(1+t^2)y=t-arctan(t),求d^3y/dx^3
问一道参数方程阶导数的题目,
求出以下参数方程所确定的函数的指定阶函数.
已知:
x=ln(1+t^2)
y=t-arctan(t),
求d^3y/dx^3
问一道参数方程阶导数的题目,求出以下参数方程所确定的函数的指定阶函数.已知:x=ln(1+t^2)y=t-arctan(t),求d^3y/dx^3
dy/dx=dy/dt*dt/dx=(dy/dt)/(dx/dt)=[1-1/(1+t^2)]/[2t/(1+t^2)]=t/2
d^2y/dx^2=[d(t/2)/dt]*(dt/dx)=(1/2)/[2t/(1+t^2)]=-(1+t^2)/4t
d^3y/dx^2=[d(d^2y/dx^2)/dt]*(dt/dx)=[(1+t^2)/4t]'*(1+t^2)/(2t)
=[(t^2-1)/(4t^2)]*[(1+t^2)/(2t)]=(t^4-1)/(8t^3)
dy/dx = (dy/dt)/(dx/dt) = (1-1/(1+t^2))/(2t/(1+t^2)) = t/2
d^2y/dx^2 = d(t/2)/dx = (d(t/2)/dt)/(dx/dt) = (1/2)/(2t/(1+t^2))
= (1+t^2)/(4t)
d^3y/dx^3 = d((1+t^2)/(4t))/dx = (d((1+t^2)/(4t))/dt)/(dx/dt)
= ((t^2-1)/(4t^2))/(2t/(1+t^2)) = (t^4-1)/(8t^3)
dy/dx=(dy/dt)/(dx/dt)=(1-1/(1+t^2))/(2t/(1+t^2))=A
d^2y/dx^=d(dy/dx)/dx=d(A)/dx=(d(A)/dt)/(dx/dt)=B
d^3y/dx^3 =d(B)/dx=(d(B)/dt)/(dx/dt)=C
由于本人身边连笔也没有，实在无法算，
望楼主好好算
数学这东西多算才有进步
dy/dx
=(dy/dt)/(dx/dt)
=[1-1/(1+t^2)]/[2t/(1+t^2)]
=t/2
d^2y/dx^2
=d(t/2)/dx
=[d(t/2)/dt]/(dx/dt)
=(1/2)/[2t/(1+t^2)]
=(1+t^2)/4t设为A
d^3y/dx^3
=dA/dx
=(dA/dt)/(dx/dt)
={[8t^2-4(1+t^2)]/16t^2}/[2t/(1+t^2)]
=(t^2-1)(1+t^2)/8t^3
=(t^4-1)/(8t^3)
两边同时求导