connection derivation in algebra and calculus

in algebra, definition of derivation is:

let R be ring or nearing or primering etc..

an additive mapping d:R\toR said to be derivation on R if d(xy)=d(x)y+yd(x)

additive mapping satisfy d(x+y)=d(x)+d(y)

for all x,y\inR

when i come with example, generaly given the commutator f(x)= ax-xa. of course that satisfy aditive mapping and derivation.

I want to make different example, and confuse with derivation in calculus, cause let f(x)= 2x^2+4, of course f'(x)=4x but f(x) doesn't satisfy additive maping nor derivation

is there any connection between them?

(i thin'k derivation in algebra similar to product rule in calculus, but this make me more confused)