Use the Chain Rule to find the indicated partial derivatives

Use the Chain Rule to find the indicated partial derivatives.

Solution

dM/du=(dM/dx).(dx/du)+(dM/dy).(dy/du)+(dM/dz).(dz/du) =e^(y-z^3).2v+(x.e^(y-z^3))+(x.(-3z^2).e^(y-z^3)) at u=6, v=-6, x=-72, y=12, z=0 dM/du=[-12(e^12)+(-72(e^12))]= -84e^12 dM/dv=(dM/dx).(dx/dv)+(dM/dy).(dy/dv)+(dM/dz).(dz/dv) =e^(y-z^3).2u-(x.e^(y-z^3))+(x.(-3z^2).e^(y-z^3)) at u=6, v=-6, x=-72, y=12, z=0 dM/dv=[12(e^12)-(-72(e^12))]= 84e^12 dM/du= -84e^12 dM/dv= 84e^12