Differentiation rules
Part of a series of articles about  
Calculus  





Specialized


This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Contents
 1 Elementary rules of differentiation
 2 Power laws, polynomials, quotients, and reciprocals
 3 Derivatives of exponential and logarithmic functions
 4 Derivatives of trigonometric functions
 5 Derivatives of hyperbolic functions
 6 Derivatives of special functions
 7 Derivatives of integrals
 8 Derivatives to nth order
 9 See also
 10 References
 11 Sources and further reading
 12 External links
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^{[1]}^{[2]}—including complex numbers (C).^{[3]}
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 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 with respect to is
In Leibniz's notation this is correctly 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 g(f(x)) = x and f(g(y)) = y, 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
Special cases include:
 If f(x) = x, then f′(x) = 1. This special case may be generalized to:
 The derivative of an affine function is constant: if f(x) = ax + b, then f′(x) = a.
Combining this rule with the linearity of the derivative and the addition rule permits the computation of the derivative of any polynomial.
The reciprocal rule
The derivative of h(x) = 1/f(x) for any (nonvanishing) function f is:
In Leibniz's notation, this is written
The reciprocal rule can be derived from the quotient rule.
The quotient rule
If f and g are functions, then:
 wherever g is nonzero.
This can be derived from the product 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,
wherever both sides are well defined.
Special cases:
 If f(x) = x^{a}, f′(x) = ax^{a − 1} when a is any nonzero real number and x is positive.
 The reciprocal rule may be derived as the special case where g(x) = −1.
Derivatives of exponential and logarithmic functions
note that the equation above is true for all c, but the derivative for c < 0 yields a complex number.
the equation above is also true for all c but yields a complex number if c<0.
Logarithmic derivatives
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. 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
, and 
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 :
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.
Derivatives to nth order
Some rules exist for computing the nth 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 nonnegative integer solutions of the Diophantine equation .
General Leibniz rule
If f and g are n times differentiable, then
See also
 Vector calculus identities
 Differentiable function
 Differential of a function
 List of mathematical functions
 Trigonometric functions
 Inverse trigonometric functions
 Hyperbolic functions
 Inverse hyperbolic functions
 Matrix calculus
 Differentiation under the integral sign
References
 ^ Calculus (5th edition), F. Ayres, E. Mendelson, Schaum's Outline Series, 2009, ISBN 9780071508612.
 ^ Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, ISBN 9780071623667.
 ^ Complex Variables, M.R. Speigel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 9780071615693
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 9780071548557.
 The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 9780521575072.
 Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 9780521861533
 NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010, ISBN 9780521192255.
External links
Library resources about Differentiation rules 
 Derivative calculator with formula simplification