解答
dtd((w(t,x(w,t),y(w,t),z(w,t)))(x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2))
解答
w((,,,(1⋅x(w,t)y(w,t)z(w,t)+(dtd(x(w,t))y(w,t)z(w,t)+(dtd(y(w,t))z(w,t)+dtd(z(w,t))y(w,t))x(w,t))t))(x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2)+((dtd(x(w,t))(w,t)2+2(w,t)(w,1)x(w,t))+(dtd(y(w,t))(w,t)2+2(w,t)(w,1)y(w,t))+(dtd(z(w,t))(w,t)2+2(w,t)(w,1)z(w,t)))(t,x(w,t),y(w,t),z(w,t)))
求解步骤
dtd((w(t,x(w,t),y(w,t),z(w,t)))(x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2))
将 w视为常数
将常数提出: (a⋅f)′=a⋅f′=wdtd((t,x(w,t),y(w,t),z(w,t))(x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2))
使用微分乘法定则: (f⋅g)′=f′⋅g+f⋅g′f=t,x(w,t),y(w,t),z(w,t),g=x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2=w(dtd(t,x(w,t),y(w,t),z(w,t))(x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2)+dtd(x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2)(t,x(w,t),y(w,t),z(w,t)))
dtd(t,x(w,t),y(w,t),z(w,t))=(,,,(1⋅x(w,t)y(w,t)z(w,t)+(dtd(x(w,t))y(w,t)z(w,t)+(dtd(y(w,t))z(w,t)+dtd(z(w,t))y(w,t))x(w,t))t))
dtd(x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2)=(dtd(x(w,t))(w,t)2+2(w,t)(w,1)x(w,t))+(dtd(y(w,t))(w,t)2+2(w,t)(w,1)y(w,t))+(dtd(z(w,t))(w,t)2+2(w,t)(w,1)z(w,t))
=w((,,,(1⋅x(w,t)y(w,t)z(w,t)+(dtd(x(w,t))y(w,t)z(w,t)+(dtd(y(w,t))z(w,t)+dtd(z(w,t))y(w,t))x(w,t))t))(x(w,t)(w,t)2+y(w,t)(w,t)2+z(w,t)(w,t)2)+((dtd(x(w,t))(w,t)2+2(w,t)(w,1)x(w,t))+(dtd(y(w,t))(w,t)2+2(w,t)(w,1)y(w,t))+(dtd(z(w,t))(w,t)2+2(w,t)(w,1)z(w,t)))(t,x(w,t),y(w,t),z(w,t)))