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Elementary rules of differentiation
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).
Differentiation is linear
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
The 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
The 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:
The inverse function rule
If the function f has an inverse function g, meaning that and , then
In Leibniz notation, this is written as
Power laws, polynomials, quotients, and reciprocals
The polynomial or elementary 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.
The 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.
The 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.
Generalized power rule
The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g,
- If , then when a is any non-zero real number and x is positive.
- The reciprocal rule may be derived as the special case where .
Derivatives of exponential and logarithmic functions
the equation above is true for all c, but the derivative for yields a complex number.
the equation above is also true for all c, but yields a complex number if .
- wherever f is positive.
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for taking derivatives.
Derivatives of trigonometric functions
It is common to additionally define an inverse tangent function with two arguments, . Its value lies in the range and reflects the quadrant of the point . For the first and fourth quadrant (i.e. ) one has . Its partial derivatives are
Derivatives of hyperbolic functions
Derivatives of special functions
with being the digamma function, expressed by the parenthesized expression to the right of in the line above.
Derivatives of integrals
Suppose that it is required to differentiate with respect to x the function
where the functions and are both continuous in both and in some region of the plane, including , and the functions and are both continuous and both have continuous derivatives for . Then for :
Derivatives to nth order
Some rules exist for computing the n-th derivative of functions, where n is a positive integer. These include:
Faà di Bruno's formula
If f and g are n-times differentiable, then
where and the set consists of all non-negative integer solutions of the Diophantine equation .
General Leibniz rule
If f and g are n-times differentiable, then
- Calculus (5th edition), F. Ayres, E. Mendelson, Schaum's Outline Series, 2009, ISBN 978-0-07-150861-2.
- Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, ISBN 978-0-07-162366-7.
- Complex Variables, M.R. Speigel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 978-0-07-161569-3
- "The Exponent Rule for Derivatives". Math Vault. 2016-05-21. Retrieved 2019-07-25.
Sources and further reading
These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. Those in this article (in addition to the above references) can be found in:
- Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schaum's Outline Series, 2009, ISBN 978-0-07-154855-7.
- The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
- Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
- NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010, ISBN 978-0-521-19225-5.
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