We can derive the logarithmic derivative
$$\frac{\partial }{\partial t}\left\{\log(f(t))\right\} = \frac{f'(t)}{f(t)}$$
But can we similarly find some other function, so that:
$$\frac{\partial }{\partial t}\left\{h(f(t))\right\} = \frac{f''(t)}{f(t)}$$
If it is impossible, how can we show it being impossible?
Own work I first imagined something like... $$h: f\to \frac{\partial \log(f'(t)) }{\partial t} \cdot \frac{\partial \log(f(t))}{\partial t} = \frac{f''(t)}{f(t)}$$
Now this is hardly a function in any normal sense, is it?
If it is not a function, then what to call it?
If $h$ is a function such that $$\frac{\partial }{\partial t}\left\{h(f(t))\right\} = \frac{f''(t)}{f(t)}$$ holds for all $f$ then take $f(t)=t$ to see that $$\frac{\partial }{\partial t}\left\{h(t))\right\} =0$$ which means that $h$ is a constant. Hence there is no such function.