6.11 Solve implicit differentiation of equations

Suppose we are given \(z=x e^{-y}, x=\cosh (t), y=\cos (s)\) and need to find \(\frac {dz}{ds}\)

x[t_] := Cosh[t] y[s_] := Cos[s] z[x_, y_] := x[t] Exp[-y[s]] D[z[x, y], s]

(Cosh[t]*Sin[s])/E^Cos[s]

Another example: \(u=x^2 y^3 z, x=\sin (s+t), y=\cos (s+t), z=e^{s t}\) and we need to find \(\frac {du}{ds}\) and \(\frac {du}{dt}\)

x[s_, t_] := Sin[s + t]; y[s_, t_] := Cos[s + t]; z[s_, t_] := Exp[s t]; u[s_, t_] := x[s, t]^2 y[s, t]^3 z[s, t]; Clear[s, t]; D[u[s, t], s]

2*E^(s*t)*Cos[s + t]^4*Sin[s + t] + E^(s*t)*t*Cos[s + t]^3*Sin[s + t]^2 - 3*E^(s*t)*Cos[s + t]^2*Sin[s + t]^3