How to calculate this derivate using the chain rule

Question

How can we get this result, step by step

$$\frac{d(\frac{f(k_t)}{k_t})}{dt}= \frac{\dot{k_t}}{k_t}(f'(k_t)-\frac{f(k_t)}{k_t})$$

Answer 1

$$\frac{d}{dt}\frac{f}{k}\stackrel{\stackrel{\text{Chain}}{\text{rule}}}{=}\dot{k}\frac{d}{dk}\frac{f}{k}\stackrel{\stackrel{\text{Product}}{\text{rule}}}{=}\dot{k}\left(\frac{f^\prime}{k}-\frac{f}{k^2}\right)=\frac{\dot{k}}{k}\left(f^\prime-\frac{f}{k}\right).$$