Using the derivative notation, we say

$$ {\operatorname{d} \over \operatorname{d}\!x} \left [  x^2 + 1  \right ]   =   2 x $$

Using the differential notation, we say the same thing as:

$$ d \left [  x^2 + 1  \right ]   =   2 { \operatorname{d}\!x} $$

Differential notation has the same rules as derivatives:

$$ {\operatorname{d} \over \operatorname{d}\!x} \left [  c  \right ]   =   0                   \Rightarrow        \operatorname{d} [ c ] = 0 $$

$$ {\operatorname{d} \over \operatorname{d}\!x} \left [  c  \operatorname{d}\!f  \right ]   =   c  {\operatorname{d}\!f \over \operatorname{d}\!x}                   \Rightarrow        \operatorname{d} [ c \operatorname{d}\!f ]  =  c \operatorname{d}\!f $$

$$ {\operatorname{d} \over \operatorname{d}\!x} \left [  f + g  \right ]   =   {\operatorname{d}\!f \over \operatorname{d}\!x} + {\operatorname{d}\!g \over \operatorname{d}\!x}                     \Rightarrow        \operatorname{d} [ f + g ]  =  \operatorname{d}\!f  + \operatorname{d}\!g $$

... and so on.

Note that a derivative deals with the tangent line to a function. Differentials deal with the actual function. In other words, the derivative measures the rate of change, while the differential measures the change itself.

If \( y \) is distance and \( x \) is time, then \( {\operatorname{d}\!y \over \operatorname{d}\!x} \) is measured in distance over time, which is speed. But \( \operatorname{d}\!y \) is measured in distance, because it is a change in distance.

Sometimes you want to know how fast something is changing. That's when you use the derivative. Sometimes you want to know how much something has changed. Then you use the differential.