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