Chain Rule Versions of the Basic Derivative Rules

Here we summarize the chain rule versions of all of the basic rules you should know for finding derivatives of functions. If the goal were to be able to efficiently calculate the derivative of a function then you should memorize all of these rules. However, in order to fully understand derivatives you should also know how these rules are related and how to derive some of the rules from others. In particular you should not forget that you are using the chain rule whenever you apply one of the following rules.

Note that u represents a function of x or, in general, a function of the variable the derivative is being taken with respect to. The derivative of u with respect to x is u′. If you compare these derivatives to those of the basic functions you will see they are the same except:

1. the argument of the basic function is now another function u, which appears unchanged in the derivative, instead of just x; and
2. the derivative of the basic function is multiplied by the derivative of this other function.