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Concept
Differentiation rules
Summary
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.
Official source
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