Question

Let z(s,t) z(s,t) be a differentiable function of two variable s and t with continuous first partial derivatives. Assume further that s=s(x,y), t=t(x,y) are differentiable functions of variables x and y .

(a) find the general chain rule chain rule for (∂z/∂x)y . Your subscripts will be marked.

(b) Let equations

F(x,y,s,t)=y+x^2+cos(t)−sin(s)+1=0,

G(x,y,s,t)=x+y^2−st−2=0,

implicitly define s , and t as functions of x and y .

Compute ∂s/∂x)y ∂t/∂x)y at the point P(x,y,s,t)=(1,−1,π,0).

(c) Let now z(s,t)=s^2+t. By using the above two parts evaluate ∂z/∂x)y at P defined in part (b).

Answer 1