formula transformation derivative trick

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I have the function f(t). Is there any reshaping trick to get from $\frac{1}{2}\frac{f'^2}{f^2}$ to $\frac{f''}{f}$

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You can differentiate $ \dfrac {f'}f$: $$\left (\dfrac {f'}f\right)'=\dfrac {f''}{f}-\dfrac {f'^2}{f^2}$$