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二阶导数怎么求?
2022-11-16
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dx、dy表示微分,当然可以拆开,
对于参数方程,x=f(t),y=g(t),
对于参数方程,先求微分:dx=f'(t)dt,dy=g'(t)dt,
dy/dx=g'(t)/f'(t),
而如果先消去参数,t=fˉ¹(x),y=g(fˉ¹(x))
dy/dx=g'(fˉ¹(x))*fˉ¹'(x)=g'(fˉ¹(x))/f'(t)=g'(t)/f'(t),是一样的。
而二阶导数,注意是d²y/dx²
是什么意思呢?就是这里要把dy/dx看成是新的“y”,x还是等于f(t),
所以应该这样:d(dy/dx)=[g'(t)/f'(t)]'dt=[g''(t)f'(t)-g'(t)f''(t)]/f'(t)² dt
dx=f'(t)dt
d²y/dx²=d(dy/dx)/dx=[g''(t)f'(t)-g'(t)f''(t)]/f'(t)³
对于参数方程,x=f(t),y=g(t),
对于参数方程,先求微分:dx=f'(t)dt,dy=g'(t)dt,
dy/dx=g'(t)/f'(t),
而如果先消去参数,t=fˉ¹(x),y=g(fˉ¹(x))
dy/dx=g'(fˉ¹(x))*fˉ¹'(x)=g'(fˉ¹(x))/f'(t)=g'(t)/f'(t),是一样的。
而二阶导数,注意是d²y/dx²
是什么意思呢?就是这里要把dy/dx看成是新的“y”,x还是等于f(t),
所以应该这样:d(dy/dx)=[g'(t)/f'(t)]'dt=[g''(t)f'(t)-g'(t)f''(t)]/f'(t)² dt
dx=f'(t)dt
d²y/dx²=d(dy/dx)/dx=[g''(t)f'(t)-g'(t)f''(t)]/f'(t)³
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