Examples of the Chain Rule
Andy Long
Calculus I

Examples:

    #9, p. 154
    #16
    #29
    #42
    #77

Clear["Global`*"]

#9

F[x_]:=Sqrt[1-2x]
F'[x]
(* Now let's identify the composition: *)
f[x_]:=Sqrt[x]
g[x_]:=1-2x
f'[g[x]]g'[x] (* same thing! Thank goodness.... *)

#16

Clear[n]
F[x_]:=6 Cot[n x]
F'[x]
(* Now let's identify the composition: *)
f[x_]:=6 Cot[x]
g[x_]:=n x
f'[g[x]]g'[x] (* same thing! Thank goodness.... *)

#29: this is an interesting example of more than two functions in a chain....

F[x_]:=Sin[Sqrt[1+x^2]]
F'[x] (* = f(g(h(x))) *)
(* Now let's identify the composition: three functions this time! *)
f[x_]:=Sin[x]
g[x_]:=Sqrt[x]
h[x_]:=1+x^2
f'[g[h[x]]]g'[h[x]]h'[x] (* same thing! Thank goodness.... *)

#77

B[t_]:=4.0+0.35Sin[2Pi t/5.4]
B'[t]
B'[1]
(* Now let's identify the composition: *)
f[x_]:=4.0+0.35Sin[x]
g[t_]:=2Pi t/5.4
f'[g[t]]g'[t] (* same thing! Thank goodnes.... *) 
f'[g[1]]g'[1]