Parameterized derivative when chain rule usually in order

Question

I have the following problem, which requires the chain rule and can be a little tedious. I'm trying to figure out how to do this in Mathematica, but I can't get a solution. I know the answer, which is 20.085. Here's the problem:

And here's my code:

Clear[x, y, f, t]

x[t_] := Sin[t];

y[t_] := Cos[t]

z[t_] := 2 t^3 + 3

f[t_] := x[t] E^(y[t] z[t])

f'[t]

And Mathematica outputs the following:

E^((3 + 2 t^3) Cos[t]) Cos[t] + 
 E^((3 + 2 t^3) Cos[t]) Sin[t] (6 t^2 Cos[t] - (3 + 2 t^3) Sin[t])

Any thought on how I can structure some code that can be used to solve other of these parameterized functions that would ordinarily require the chain rule?

Thanks so much.

Answer 1

w = x*E^(y*z);
x = Sin[t];
y = Cos[t];
z = 2 t^3 + 3;
D[w, t] /. t -> 0
% // N

E^3

20.0855

Answer 2

Clear[x, y, f, t]
x[t_] = Sin[t];
y[t_] = Cos[t]
z[t_] = 2 t^3 + 3
f[t_] = x[t] E^(y[t] z[t])
f'[t]
% /. t -> 0