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I am familiar with two types of differential

normal differential: $$d^2x^a$$

covariant differential: $${\mathcal D^2x^a}=d^2x^a+\Gamma^a_{bc}dx^bdx^c$$ (where the covariant differential is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component)

Is there any other kind of differential?

Are there more types of differential in mathematics?

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Yes. See en.wikipedia.org/wiki/Differential_%28mathematics%29 But rather than learning about "things with the same name", it is better to ask about "things of the same nature". What is it that you know about the "normal" and "covariant" differentials that makes you think about other ones? –  Willie Wong Jan 8 '13 at 16:28
    
@Willie Wong I am very interested in differential in these days –  Neo Jan 8 '13 at 16:43
    
A ring derivation in abstract algebra (see Encyclopedia of Mathematics entry for Derivation in a ring and the Wikipedia article on Differential Algebra) might qualify for what you're asking about and it's a bit more removed from the elementary calculus notion than the examples you gave. –  Dave L. Renfro Jan 8 '13 at 16:54
    
Ah, I see now that you could have learned about ring derivations from the Wikipedia article that Willie Wong cited (because that article cites the Wikipedia article on Differential Algebra). –  Dave L. Renfro Jan 8 '13 at 17:11
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