Elementary derivatives

In order to be able to differentiate efficiently, one needs to know derivatives of elementary functions. Here comes the list:

You should know all of them by heart (in some courses they skip hyperbolic and related functions, which would shorten the list a bit). Some functions appear so often that, although they are in fact just special cases of the above, it is worthwile to remember their derivatives as well.

They are all based on the "power" rule:

The fact about [Ax + B]′ also uses linearity of derivative (see the section Derivative and operations).

Remark: One can show that the rule for logarithm can be extended. We can say that the derivative of ln|x| is 1/x for all non-zero x. This is extremely useful when it comes to integration, see Elementary integrals in Integrals - Theory - Methods of Integration, but like us most authors prefer the simpler version in their list of elementary derivatives. Note that for handling absolute value we have a general trick that helps not only with logarithm (see Methods Survey - Calculating Derivative).

Bonus: Beware! A mathematical joke coming!
Here's one of my favorite math jokes to help you remember one of the facts above. In a psychiatric ward one patient runs around scaring other patients by threatening to differentiate them. Of course, the other patients are really scared, running away and trying to hide, just one of them seems rather indifferent. This is really hard on the "differentiator", so he comes hard at the indifferent one: "Boo boo boo, I'll differentiate you!" But the other remains calm, hardly pays any attention. "Aren't you scared?" asks the differentiator. "No, I am e to x."

Higher order derivative
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