Final answer:
To find dz/dt using the Chain Rule, we need to find the partial derivatives of z with respect to x and y, and then find dx/dt and dy/dt. Finally, we can apply the Chain Rule to find dz/dt.
Step-by-step explanation:
To find dz/dt using the Chain Rule, we can start by finding the partial derivatives of z with respect to x and y. Then we can use the given expressions for x and y to find dx/dt and dy/dt. Finally, we can apply the Chain Rule to find dz/dt.
First, let's find the partial derivatives of z:
∂z/∂x = cos(x)cos(y)
∂z/∂y = -sin(x)sin(y)
Next, let's find dx/dt and dy/dt:
dx/dt = (1/2√t)
dy/dt = -4/t^2
Finally, we can apply the Chain Rule:
dz/dt = (∂z/∂x) * (dx/dt) + (∂z/∂y) * (dy/dt)
Substituting the partial derivatives and the expressions for dx/dt and dy/dt:
dz/dt = (cos(x)cos(y)) * (1/2√t) + (-sin(x)sin(y)) * (-4/t^2)
Simplifying further:
dz/dt = (cos(√t)cos(4/t)) * (1/2√t) + (-sin(√t)sin(4/t)) * (-4/t^2)
So, dz/dt = (cos(√t)cos(4/t)) * (1/2√t) + (sin(√t)sin(4/t)) * (4/t^2)