Final answer:
The student asked to find dz/dt using the Chain Rule for the function z = sin(x) cos(y), with x and y as functions of t. Calculus and Chain Rule are used to differentiate the given functions and find the time derivative of z.
Step-by-step explanation:
To find dz/dt using the Chain Rule for the function z = sin(x) cos(y), where x = sqrt(t) and y = 4/t, we will differentiate z with respect to x and y and then with respect to t.
The partial derivatives of z are:
- dz/dx = cos(x) cos(y)
- dz/dy = -sin(x) sin(y)
Then using the x(t) and y(t) given:
- dx/dt = 1/(2 sqrt(t))
- dy/dt = -4/t^2
Applying the Chain Rule to find dz/dt:
dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)
Substitute the partial derivatives and the derivatives of x and y with respect to t:
dz/dt = cos(x) cos(y)*(1/(2 sqrt(t))) - sin(x) sin(y)*(-4/t^2)
Finally, substitute x and y back into the equation to get dz/dt in terms of t.