Calculus

Think in terms of X-Rays and Time-lapse - betterexplained.com

Derivative

ddxy\frac{d}{dx} y

This is X-Ray, which means split the whole into (possibly different) parts along the x-axis

Think of a circle: if you split the circle into smaller circles so that each sub-circle has some smaller radius than the original one then it's called X-Ray vision.

X-Ray is not a concept in Calculas. It's simply used just to describe it.

Integral

∫r=0r2πr dr\int_{r=0}^{r} 2 \pi r \,dr

This is Time-Lapse, which means accumulate (possibly different) steps along the r direction