Final answer:
The partial derivatives of z with respect to s and t are found using the Chain Rule, where ∂z/∂s = z/t - 7zt/s^2 and ∂z/∂t = -zs/t^2 + 7z/s.
Step-by-step explanation:
The student has asked to use the Chain Rule to find the partial derivatives ∂z/∂s and ∂z/∂t of the function z = ex+7y, given that x = s/t and y = t/s. To find these derivatives, we apply the Chain Rule, which tells us how to differentiate composite functions.
First, we find ∂z/∂s:
∂z/∂x = ex+7y = z (since z is the function ex+7y)
∂x/∂s = 1/t
∂z/∂y = 7z (because the derivative of ex+7y with respect to y is 7ex+7y)
∂y/∂s = -t/s2
Using the Chain Rule, we get:
∂z/∂s = ∂z/∂x ∗ ∂x/∂s + ∂z/∂y ∗ ∂y/∂s
∂z/∂s = z/t - 7zt/s2
Next, we find ∂z/∂t:
∂z/∂x = z
∂x/∂t = -s/t2
∂z/∂y = 7z
∂y/∂t = 1/s
Applying the Chain Rule again, we obtain:
∂z/∂t = ∂z/∂x ∗ ∂x/∂t + ∂z/∂y ∗ ∂y/∂t
∂z/∂t = -zs/t2 + 7z/s