Differentiation rules

  (Redirected from Constant multiple rule)

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]

Differentiation is linearEdit

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 ruleEdit

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  , this is written in a more concise way as:

 

The inverse function ruleEdit

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 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.[4]

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  .

 
 
 
 
   

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. 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

   
   
   
   
   
   

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 functionsEdit

   
   
   
   
   
   

Derivatives of special functionsEdit

Gamma function  
 
 
 

with   being the digamma function, expressed by the parenthesized expression to the right of   in the line above.

Riemann Zeta function 
 
 

Derivatives of integralsEdit

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 orderEdit

Some rules exist for computing the n-th derivative of functions, where n is a positive integer. These include:

Faà di Bruno's formulaEdit

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 ruleEdit

If f and g are n-times differentiable, then

 

See alsoEdit

ReferencesEdit

  1. ^ Calculus (5th edition), F. Ayres, E. Mendelson, Schaum's Outline Series, 2009, ISBN 978-0-07-150861-2.
  2. ^ Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, ISBN 978-0-07-162366-7.
  3. ^ 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
  4. ^ "The Exponent Rule for Derivatives". Math Vault. 2016-05-21. Retrieved 2019-07-25.

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.

External linksEdit