Answer
$\frac{dy}{dx} = \frac{1}{4}e^{\frac{-3θ}{2}}$
Work Step by Step
1. Find $\frac{dy}{d θ }$:
$y = e^{-\frac{θ}{2}}$
**$ u = -\frac{θ}{2} $
**$du = -\frac{1}{2}dθ$
$y = e^u$
$dy = (e^u)du$
$dy = e^{-\frac{θ}{2}} * (-1/2)d θ $
$dy/d θ = \frac{e^{-\frac{θ}{2}}}{2}$
2. Find $\frac{dx}{d θ }$:
$x= 2e^θ $
$dx = 2(e^θ)dθ$
$\frac{dx}{d θ } = 2e^θ $
3. Using the expression: $\frac{dy}{dx} = \frac{dy/d θ }{dx/d θ }$, find the derivative:
$\frac{dy}{dx} = \frac{\frac{e^{-\frac{θ}{2}}}{2}}{2e^θ} = \frac{e^{-\frac{θ}{2}}}{4e^θ} = \frac{1}{4}e^{\frac{-3θ}{2}}$
