Why is the derivative of $x^2$ equal to $2x$ rather than $4x$?

Question

Between 2m26s and 4m36s of this YouTube video, the teacher explains the power rule using the area of a square.

1. But why isnt the square or cube increasing its area or volume on all sides?

2. If square increases its area by all 4sides, shouldnt the derivative be 4x dx ?? whats the argument behind this???

Thanks.

Answer 1

That's because the core thing which is happening is not that the square / cube is padded on one side, but that the side lengths of the square / cube are increasing by $dx$. We could have it expand in both directions, but then each padding would be $\frac12dx$ thick. You would get the same result.