r=e^θ
using polar coordinates
x=rcos(θ)=e^θcosθ
y=rsin(θ)=e^θsinθ
Horizontal tangents occur where dy/dx=0 and vertical tangents occur where
dy/dx→∞⟹dx→0.
dy/dx=dy/dθ/dx/dθ
dy/dx=e^θcosθ+e^θsinθ/−e^θsinθ+e^θcosθ
So horizontal tangents occur where
e^θcosθ+e^θsinθ=0⟹θ=−π/4
Vertical tangents occur where
−e^θsinθ+e^θcosθ=0⟹θ=π/4
