Use chain rule to find dz/dt.
z=tan^-1 (y/x) , x=e^t, y=7-e^(-t)
We begin by finding the partial derivatives of z with respect to x and y:
∂z/∂x = -y/(x^2 + y^2)
∂z/∂y = x/(x^2 + y^2)
Next, we use the chain rule to find dz/dt:
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
Substituting in the given values of x and y and their derivatives:
dz/dt = (-[7-e^(-t)]/[e^(2t) + (7-e^(-t))^2])(e^t) + (e^t/[e^(2t) + (7-e^(-t))^2])(e^(-t))
Simplifying, we get:
dz/dt = [-7e^t + 2e^(-t)]/[e^(2t) + (7-e^(-t))^2]