# Can we interchange derivatives when multiplied?

Although this looks like a physics question, this is more of a Math question, I was reading the Energy-Mass relationship derivation, it goes as follows,

Force $F$ is given by

$$F=\frac{d}{dt}(mv)$$ $$=\frac{dm}{dt}v+m\frac{dv}{dt}$$

And kinetic energy $K$ $$dK=F \cdot ds$$ So, $$dK=\frac{dm}{dt}v\cdot ds+m\frac{dv}{dt} \cdot ds$$

The next step what I don't understand

$$dK=dm \cdot v \cdot \frac{ds}{dt}+m \cdot dv \cdot \frac{ds}{dt}$$

Can we really write something like for example $$\bigg(\frac{dx}{dt}dy \bigg) \space \text{as} \space \bigg (dx \frac{dy}{dt} \bigg)?$$ I dont think that is valid, or am I missing out something?

Should I continue reading it or throw away the book?

• That does look like something physicists would do, yes. – hmakholm left over Monica Mar 25 '16 at 15:22
• you mean its not valid in the strict sense? can that still be done? – Courage Mar 25 '16 at 15:24

$$\frac{dx}{dt}dy=\frac{dx}{dt}\frac{dy}{dx}dx=\frac{dy}{dt}dx$$