x^(2)+y^(2)+z^(2)=9
Differentiate both sides with respect to t
(d(x^(2)+y^(2)+z^(2)))/(dt)=(d(9))/(dt)
(d(x^(2)))/(dx)+(d(y^(2)))/(dt)+(d(z^(2)))/(dt)=0
Use chain rule as shown below:
(d(x^(2)))/(dx)* (dx)/(dt)+(d(y^(2)))/(dy)* (dy)/(dt)+(d(z^(2)))/(dz)* (dz)/(dt)=0
(2x)* (dx)/(dt)+(2y)* (dy)/(dt)+(2z)* (dz)/(dt)=0
Divide both sides by 2
(x)* (dx)/(dt)+(y)* (dy)/(dt)+(z)* (dz)/(dt)=0
Substitute x=2, y=2, z=1 and (dx)/(dt)=5, (dy)/(dt)=4
(2)* (5)+(2)* (4)+(1)* (dz)/(dt)=0
10+8+(dz)/(dt)=0
(dz)/(dt)=-18
Result:
(dz)/(dt)=-18