Let $f(x)=x^x$ and $g(x)=\Big(\frac{1}{x}\Big)^{x}$ .Let $A$ be the tangent of $f(x)$ at $x=1$ and $B$ the tangent of $g(x)$ at $x=1$ . Then A and B are perpendicular .
Proof without first derivative
Clearly since the equation $x^x=x$ have a unique solution which is $x=1$ we conclude that $y=x$ is a tangent of the function $f(x)$ using convexity and the three chord lemma .The same reasoning apply to the equation $\Big(\frac{1}{x}\Big)^{x}=2-x$ and using convexity conducts to $y=2-x$ is a tangent of the function $g(x)$ . Now it's easy to conclude .
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Let $f=h$ and $g=1/h$. Then $f=g\iff h^2(x)=1$. Furthermore, $$f'\cdot g'=h'\cdot(-1)\cdot\frac{1}{h^2}h'.$$ Thus the tangents in the point of intersection are perpendicular iff $(h'(x))^2=1.$