Differentiation rules

Summary

This article is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

Elementary rules of differentiation

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Unless otherwise stated, all functions are functions of real numbers ( ) that return real values, although, more generally, the formulas below apply wherever they are well defined,[1][2] including the case of complex numbers ( ).[3]

Constant term rule

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For any value of  , where  , if   is the constant function given by  , then  .[4]

Proof

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Let   and  . By the definition of the derivative:  

This computation shows that the derivative of any constant function is 0.

Intuitive (geometric) explanation

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The derivative of the function at a point is the slope of the line tangent to the curve at the point. The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0.

In other words, the value of the constant function,  , will not change as the value of   increases or decreases.

 
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 0 where black and solid.

Differentiation is linear

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For any functions   and   and any real numbers   and  , the derivative of the function   with respect to   is  .

In Leibniz's notation, this formula is written as:  

Special cases include:

  • The constant factor rule:

 

  • The sum rule:

 

  • The difference rule:

 

Product rule

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For the functions   and  , the derivative of the function   with respect to   is:  

In Leibniz's notation, this formula is written:  

Chain rule

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The derivative of the function   is:  

In Leibniz's notation, this formula is written as:   often abridged to:  

Focusing on the notion of maps, and the differential being a map  , this formula is written in a more concise way as:  

Inverse function rule

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If the function   has an inverse function  , meaning that   and  , then:  

In Leibniz notation, this formula is written as:  

Power laws, polynomials, quotients, and reciprocals

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Polynomial or elementary power rule

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If  , for any real number  , then:  

When  , this formula 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

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The derivative of   for any (nonvanishing) function   is:   wherever   is nonzero.

In Leibniz's notation, this formula 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

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If   and   are functions, then:   wherever   is nonzero.

This can be derived from the product rule and the reciprocal rule.

Generalized power rule

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The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions   and  ,   wherever both sides are well defined.

Special cases:

  • If  , then   when   is any nonzero real number and   is positive.
  • The reciprocal rule may be derived as the special case where  .

Derivatives of exponential and logarithmic functions

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  The equation above is true for all  , but the derivative for   yields a complex number.

 

  The equation above is also true for all   but yields a complex number if  .

 

 

  where   is the Lambert W function.

 

 

 

Logarithmic derivatives

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The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):   wherever   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 functions

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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:  

Derivatives of hyperbolic functions

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Derivatives of special functions

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Gamma function

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

Riemann zeta function

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Derivatives of integrals

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Suppose that it is required to differentiate with respect to   the function:  

where the functions   and   are both continuous in both   and   in some region of the   plane, including  , where  , 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

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Some rules exist for computing the  th derivative of functions, where   is a positive integer, including:

Faà di Bruno's formula

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If   and   are  -times differentiable, then:   where   and the set   consists of all non-negative integer solutions of the Diophantine equation  .

General Leibniz rule

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If   and   are  -times differentiable, then:  

See also

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References

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  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. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 978-0-07-161569-3
  4. ^ "Differentiation Rules". University of Waterloo – CEMC Open Courseware. Retrieved 3 May 2022.

Sources and further reading

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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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  • Derivative calculator with formula simplification
  • The table of derivatives with animated proves