By solving the following expression by chain rule:
$$\dfrac{d}{d[f(s)]} \left( \dfrac{d[f(s)]}{ds} \right) = \dfrac{d[f'(s)]}{d[f(s)]} =\dfrac{d[f'(s)]}{ds}\dfrac{ds}{d[f(s)]} =\dfrac{f''(s)}{f'(s)} $$
Now by solving by Clairaut's theorem:
$$ \dfrac{d}{d[f(s)]} \left( \dfrac{d[f(s)]}{ds} \right) =\dfrac{d}{ds} \left( \dfrac{d[f(s)]}{d[f(s)]} \right) =\dfrac{d(1)}{ds}=0 $$
Why is this contradiction? What am I missing?