This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Elementary rules of differentiationEdit
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[1][2] — including the case of complex numbers (C).[3]
Constant term ruleEdit
For any value of , where , if is the constant function given by , then .[4]
ProofEdit
Let and . By the definition of the derivative,
This shows that the derivative of any constant function is 0.
Intuitive (geometric) explanationEdit
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because tangent line to the constant function is horizontal and it's angle is zero.
At each point, the derivative is the slope of a line that is tangent to the curve at that point. Note: the derivative at point A is positive where green and dash–dot, negative where red and dashed, and zero where black and solid.
Differentiation is linearEdit
For any functions and and any real numbers and , the derivative of the function with respect to is:
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 ruleEdit
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{D}}
, this is written in a more concise way as:
The inverse function ruleEdit
If the function f has an inverse functiong, meaning that and then
In Leibniz notation, this is written as
Power laws, polynomials, quotients, and reciprocalsEdit
The polynomial or elementary power ruleEdit
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 ruleEdit
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 ruleEdit
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 ruleEdit
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.
Special cases
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 functionsEdit
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 .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \frac{d}{dx}\left( W(x)\right) = {1 \over {x+e^{W(x)}}} ,\qquad x > -{1 \over e}.\qquad}
where is the Lambert W function
Logarithmic derivativesEdit
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):
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.[citation needed]
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 functionsEdit
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (\cos x)' = -\sin x = \frac{e^{-ix} - e^{ix}}{2i} }
The derivatives in the table above are for when the range of the inverse secant is and when the range of the inverse cosecant is .
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
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 :
^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. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 978-0-07-161569-3
^"Differentiation Rules". University of Waterloo – CEMC Open Courseware. Retrieved 3 May 2022.
Sources and further readingEdit
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.