大师作文网问一道参数方程阶导数的题目,
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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
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)
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)