$dy\over dx$ is one things but why in integration we can treat it as 2 different terms

Question

when i am learning differentiation, my lectuer tell us that the deriative $dy\over dx$ is one things, it is not the ration between dy and dx. However when i learn about integrating, sometime we need to do substitution, like integrating $\int_{0}^{1}2xdx$ when substituting $y=2x$, we can substitute $dy=2dx$, but why in this case it can be treated as 2 different terms instead of 1 term??

Answer 1

The reason is that $dy/dx$ is the limit of $\Delta y/\Delta x$ as $\Delta x \rightarrow 0$. The limit process allows us to cheat a bit and consider the derivative as the ratio of $\Delta y/\Delta x$. This serves our purpose for integration; when we write

$$dy = 2 dx$$

we mean

$$\Delta y = 2 \Delta x$$

and then we may take that limits as $\Delta x \rightarrow 0$.