This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined — including the case of complex numbers (C). For any value of , where , if is the constant function given by , then . Let and . By the definition of the derivative, This shows that the derivative of any constant function is 0. Linearity of differentiation For any functions and and any real numbers and , the derivative of the function with respect to is: In Leibniz's notation this is written as: Special cases include: The constant factor rule The sum rule The subtraction rule Product rule For the functions f and g, the derivative of the function h(x) = f(x) g(x) with respect to x is In Leibniz's notation this is written Chain rule The derivative of the function is In Leibniz's notation, this is written as: often abridged to Focusing on the notion of maps, and the differential being a map , this is written in a more concise way as: Inverse functions and differentiation If the function f has an inverse function g, meaning that and then In Leibniz notation, this is written as Power rule If , for any real number then When this becomes the special case that if then Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial. Reciprocal rule The derivative of for any (nonvanishing) function f is: wherever f is non-zero. In Leibniz's notation, this is written The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule. Quotient rule If f and g are functions, then: wherever g is nonzero. This can be derived from the product rule and the reciprocal rule. Power rule The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g, wherever both sides are well defined.