The derivative of any polynomial function of one variable is easily obtained. If (or a constant function) and are both differentiable on some set , then so are , , and . If, in addition, is nonzero on , then (and also ) are differentiable on . Also, if is differentiable on , then is differentiable on . For the trivial case of , for some constant (a degree 0 polynomial):

For any:

Which covers any single variable polynomial function. Derivatives of non-polynomial functions require additional rules.

For any real-valued differentiable functions :

- The definition of the derivative:

Basic derivative rules:

- (Derived from Constant Multiple rule)

- (Derived from Quotient rule)

- [Proof]

Logarithmic and exponential functions:

- [Proof]

- [Proof]

Other:

[Proof]

[Proof]