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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.

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Related concepts (9)

Differentiation rules

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.

Logarithmic derivative

In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where is the derivative of f. Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f.

Quotient rule

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is It is provable in many ways by using other derivative rules. Given , let , then using the quotient rule: The quotient rule can be used to find the derivative of as follows: Reciprocal rule The reciprocal rule is a special case of the quotient rule in which the numerator .

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