= Tangent through the origin to an exponential curve
{title2=$x_*=1/a,\quad k=ae$}
For a nonzero real <constant> $a$, the <tangent line> to $y=e^{ax}$ at $x_*$ passes through the origin precisely when $e^{ax_*}-x_*ae^{ax_*}=0$. Thus $x_*=1/a$, $y_*=e$, and its <slope> is $k=ae$. This gives the unique such <tangent line>. If $a=0$, the graph is the horizontal line $y=1$ and no <tangent line> through the origin exists. The equality between the <derivative> and the slope from the origin, $f'(x_*)=f(x_*)/x_*$, is useful for other <differentiable functions> too.
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