Examples from Section 14.5: the Chain Rule
MAT329
Andy Long, Fall, 2015

#2

f[x_,y_]:=Cos[x+4 y]
x[t_]:=5 t^4
y[t_]:=1/t
D[f[x[t],y[t]],t]

#7

f[x_,y_]:=x^2 y^3
x[s_,t_]:=s Cos[t]
y[s_,t_]:=s Sin[t]
D[f[x[s,t],y[s,t]],s]
D[f[x[s,t],y[s,t]],t]

#37

CSS[T_,D_]:=1449.2+4.6 T - 0.055 T^2+ 0.00029T^3 +  0.016 D
Plot3D[CSS[T,D],{T,8,16},{D,0,20}]

CT[T_,D_]=D[CSS[T,D],T]
CD[T_,D_]=D[CSS[T,D],D]
CT[12.5,7]
CD[12.5,7]
dTdt=-2/30
dDdt=15/30
dCdt=CT[12.5,7] dTdt + CD[12.5,7] dDdt

#38

V[r_,h_]:=Pi r^2 h/3
Vr[r_,h_]=D[ V[r,h],r]
Vh[r_,h_]=D[ V[r,h],h]
Vr[120,140]
Vh[120,140]
dVdt=Vr[120,140] 18 /10+ Vh[120,140] (-25)/10