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):

[Proof]

For any:

[Proof]

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]
• [Proof]

Other:

[Proof]

[Proof]