Symbolic Differentiation

R-based solution for symbolic differentiation. It admits user-defined function as well as function substitution in arguments of functions to be differentiated. Some symbolic simplification is part of the work.

Symbolic differentiation

The original version of this software was written in R by Andrew Clausen (clausen at in 2007.

Mark Reid (mark.reid at sent a patch, applied 21/2/2009.

In 2014, Andrew has passed the maintenance to Serguei Sokol (sokol at Since then, the software was deeply rewritten and completed.

Main new features include:

  • new derivative engine allowing simple syntaxe for differentiation rules;
  • many new functions are added to the rule table;
  • custom differentiation rules can be added by user;
  • automatic differentiation (AD) of a code with multiple assignement operators;
  • when taking derivative of a function Deriv() returns a function too. The later can be called with the same arguments as the original function;
  • can differentiate by variables stored in vectors or lists, e.g. param$theta or x[1], x[2] etc.
  • simplifications are extended to rational expressions and factorizations;
  • expression caching is enabled by default;
  • Deriv() is made the only entry point for all types of entries:
    • expression
    • language
    • function
    • right hand side of a formula
    • character string
    • plain unevaluated code
  • few unit tests were added to the package



In R session do:

> library(Deriv)
> f <- function(x, n=2) x^n+sin(n*x)     # user defined function to diffierentiate
> (df <- Deriv(f))                       # -> c(x = n * x^(n - 1) + n * cos(n * x), n = log(x) * x^n + x * cos(n * x))
> df(2, 3)                               # ->         x         n

> Deriv(expression(f(y, 3)), "y")        # -> expression(3 * y^2 + 3 * cos(3 * y))
> Deriv(~ f(y, 3), "y")                  # -> 3 * y^2 + 3 * cos(3 * y)
> y <- 2; eval(Deriv(~ f(y, 3), "y"))    # -> 14.88051

For more information and examples:

> ?Deriv


Version 3.8.5

  • 2018-06-11
  • fixed Deriv::Simplify(quote(y + c(1, 2))) (issue #18)
  • fixed Deriv::Simplify(quote(y + c(0, 1))) (issue #19)
  • fixed format1() for numerical vectors requiring different digit numbers

Version 3.8.4

  • 2018-02-15
  • fixed calls to Deriv() when f is a closure (issue #17)
  • fixed call to Deriv_() when some arguments of f are not differentiable

Version 3.8.3

  • 2017-11-21
  • fixed derivation of arguments passed as c() call (issue #15)
  • edited to push forward possibility to add custom differentiation rules (issue #16)

Version 3.8.2

  • 2017-10-16
  • fixed mixing part of returned function body and its arguments when the result is a constant vector of length > 1 (issue #14)

Version 3.8.1

  • 2017-06-13
  • fixed use of argument 'combine' (which was ignored in some cases)
  • added proceeding of renamed primitive functions (e.g. f=cos; Deriv(f)) (issue #10)

Version 3.8.0

  • 2016-11-22
  • variable reassignment is now allowed in the body of differentiated function (issue #12)
  • added new optional parameter 'combine' which can be helpful in multivariate context
  • added and rep_len() to the table of derivative definitions
  • added list() to the table of linear functions
  • fixed format1() for '{...}' language expression
  • in the manual, added NB3 remark about possible vector length issues
  • fixed some simplification cases when a constant numerical vector is a part of the expression
  • fixed automatic numerical diff. tests for and rep_len()

Version 3.7.0

  • 2016-04-05
  • added optional parameter nderiv for multiple derivatives
  • expressions with indexes other than variable of differentiation are considered as constants (fixes issue#8, reported by [email protected])
  • functions that are not in drule table and not depending on x, differentiate to 0
  • for x=NULL, fixed x guess by excluding non differentiable arguments
  • fixed presence of '...' in function argument list
  • added rep() function for differentiation
  • in Cache(), simple subindex expressions are no more cached

Version 3.6.1

  • 2016-01-19
  • added 'ifelse' function to differentiable expressions ([email protected])
  • fixed trying to differentiate arguments that should not be

Version 3.6.0

  • 2015-11-04
  • added differentiation by named components of vectors or lists
  • added deCache() call prior to Cache().
  • fixed over-simplification in composite functions (issue #6 submitted by [email protected])
  • removed qlist() usage, deprecated from v3.5.3

Version 3.5.6

  • 2015-09-23
  • fixed circular variable definitions in Cache() (issue #5 submitted by [email protected])
  • fixed simplification of "{...; const}" expressions
  • fixed NOTEs from r-devel

Version 3.5.5

  • 2015-06-11
  • fixed assignement order in Deriv_() and Cache() (issue #5 submitted by [email protected])

Version 3.5.4

  • 2015-06-09
  • added the left hand side of assignment operator to simplification cache
  • Cache() is optimized to use assignements already present in the expression
  • fixed placement of auxiliary variable in Cache()
  • AD produces better code for multiple x

Version 3.5.3

  • 2015-06-03
  • qlist() is re-exported and signaled as deprecated. It will be removed in v3.6
  • added simplification log(1+x) -> log1p(x)
  • fixed power simplification in rational expressions (issue #4 submitted by [email protected])
  • fixed unnecessary repetetive differentiation in case of higher order derivatives
  • fixed alphabetique term order in simple additions

Version 3.5.2

  • 2015-05-28
  • fixed limit of 10000 chars for variable names in scache and dsym (issue #3 submitted by [email protected])

Version 3.5.1

  • 2015-05-27
  • fixed Cache() when applied to c() expression

Version 3.5

  • 2015-05-26
  • fixed abusive capturing of error messages in Simplify() (issue #1, reported by [email protected])
  • fixed interference in dsym and scache environements in nested calls to Deriv() (issue #2, reported by [email protected])
  • the function qlist() is deprecated for rule creation. Use alist() instead (seen in issue #1)
  • added pnorm() for derivative table (based on suggestion in issue # 1)
  • minor twiks in the documentation

Version 3.4

  • 2015-05-05
  • fixed usage of sinpi() and alike in R older than 3.1.0 (reported by R-core team)
  • fixed dependence of term order on LC_COLLATE in test suite

Version 3.3

  • 2015-04-10
  • fixed qlist() export in namespace (reported by Irucka Embry)
  • fixed drule example in Deriv-package manual and some other minor typos

Version 3.2

  • 2015-04-07
  • fixed date format in DESCRIPTION
  • reference to rSymPy in the package manual

Version 3.1

  • 2015-04-07
  • new syntaxe for the rule table to simplify rule writing (chain rule is implicite, non need for quote() calls)
  • added dbinom() and dnorm() to derivative table
  • sums are ordered alphabeticaly now
  • several bugs are fixed

Version 3.0

  • 2015-03-06
  • Added automatic differentiation (AD) of a code with intermediate assignements
  • Added optional expression caching (enabled by default)
  • Added new functions to rule table: abs() and sign(); special functions (beta(), gamma(), bessel() and co.; sinpi(), cospi(), tanpi(), atan2(), log1p(), expm1()
  • Added factorization of sum expressions
  • Added optional use.D parameter
  • In manual, added an example for compound function differentiation
  • If not provided, variable names are guessed from an expression to differentiate
  • If cannot be evaluted or evaluates to numeric type, the first argument of Deriv() and Simplify() is considered as a language expression
  • In unit tests, added comparison with an estimation by central differences
  • Fixed Deriv(-) call
  • Fixed Deriv(f) when f is of type .Internal()

Version 2.0

  • 2015-02-05
  • New derivative engine suppresses a need for helper functions like neg.sin()
  • Derivative table of differetiable functions is completed by trigonometric fucntions and their inverse as well as hyperbolic functions and their inverse
  • Derivative table admits different rules for a function called with different number of arguments. E.g., log() can be differentiated when called with one or two arguments. In the latter case, the base which can be constant or variable, is different from exp(1))
  • New syntax in derivative table facilitates adding user defined functions to differentiate. The same function can have several rules of differentiation depending on the number of arguments at the call moment
  • Deriv() is now the only entry point for all expressions to derive. No more need for Deriv.function() which is suppressed
  • It is now possible to pass primitive function as first argument, e.g. Deriv(sin)
  • In addition to previously possible differentiation of functions and expressions, the argument to differentiate can be submitted as
  • character string, e.g. '"x**2"'
  • right hand side of a formula, e.g '~ x**2'
  • quote() call, e.g 'quote(x**2)'
  • Power expression can be differentiated with symbolic power, e.g. 'x**n' or 'x^n'
  • Simplifications are pushed further for rational expressions (i.e. having terms united by '*' and/or '/'), all numeric terms are explicitly calculated and identical terms in numerator and denominator are simplified, identical factors are regrouped in a power term.
  • Simplifications are pushed further for linear combinations where identical terms are grouped in one term and their numerical coefficients are summed.
  • Added some simplifications for log(), exp(), power function, sqrt() and abs()
  • Some unit tests are added for Deriv() and Simplfy() (based on testthat package)

Version 1.0

  • 2014-12-10
  • Andrew Clausen passed the maintenance to Serguei Sokol
  • Serguei Sokol has fixed a bug in simplification of "a-b"
  • revamped the documentation and code as an R package
  • put the code to GitHub
  • submitted the package to CRAN

Reference manual

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3.8.5 by Serguei Sokol, 8 months ago

Report a bug at

Browse source code at

Authors: Andrew Clausen [aut] , Serguei Sokol [aut, cre]

Documentation:   PDF Manual  

Task views: Numerical Mathematics

GPL (>= 3) license

Suggests testthat

Imported by RGCCA, Sim.DiffProc, drugCombo, mirt, nCopula, neuralnet.

Depended on by ordinalCont.

Suggested by docopulae, interp, nlsr.

See at CRAN