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Symbolic differentiation, integration, series operations, limits, and transforms

Using Symbolic Math Toolbox™, you can differentiate and integrate symbolic expressions, perform series expansions, find transforms of symbolic expressions, and perform vector calculus operations by using the listed functions.

When modeling your problem, use assumptions to return the right results. See Use Assumptions on Symbolic Variables. To simplify your results, see Simplify Symbolic Expressions.


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limitLimit of symbolic expression
diffDifferentiate symbolic expression or function
functionalDerivativeFunctional derivative (variational derivative)
intDefinite and indefinite integrals
vpaintegralNumerical integration using variable precision
changeIntegrationVariableIntegration by substitution
integrateByPartsIntegration by parts
releaseEvaluate integrals
curlCurl of vector field
divergenceDivergence of vector field
gradientGradient vector of scalar function
hessianHessian matrix of scalar function
jacobianJacobian matrix
laplacianLaplacian of scalar function
potentialPotential of vector field
vectorPotentialVector potential of vector field

Series Expansions

padePade approximant
rsumsInteractive evaluation of Riemann sums
seriesPuiseux series
taylorTaylor series

Sums & Products

cumprodSymbolic cumulative product
cumsumSymbolic cumulative sum
symprodProduct of series
symsumSum of series
vpasumNumerical summation using variable precision
fourierFourier transform
ifourierInverse Fourier transform
htransHilbert transform
ihtransInverse Hilbert transform
laplaceLaplace transform
ilaplaceInverse Laplace transform
iztransInverse Z-transform
symprefSet symbolic preferences
pdeCoefficientsExtract coefficients of partial differential equation
pdeCoefficientsToDoubleConvert symbolic PDE coefficients to double format
funtoolFunction calculator
rsumsInteractive evaluation of Riemann sums
taylortoolTaylor series calculator



Differentiate symbolic expressions and functions.

Create Symbolic Functions

Use symbolic functions that accept symbolic inputs, such as f(x,y).


Integrate symbolic expressions and functions.

Taylor Series

Taylor series expansion of symbolic expressions and functions.

Fourier and Inverse Fourier Transforms

Fourier and inverse Fourier transforms of symbolic expressions.

Solve Differential Equations Using Laplace Transform

Laplace and Inverse Laplace transforms of symbolic expressions and functions.

Solve Difference Equations Using Z-Transform

Z-Transforms and inverses of symbolic expressions and functions.

Symbolic Summation

Sum symbolic vectors, matrices, or symbolic series.

Padé Approximant

Pade approximant of symbolic expressions and functions.


Limits of symbolic expressions and functions.

Find Asymptotes, Critical, and Inflection Points

Find minima, maxima, and asymptotes by using derivatives and limits.

Functional Derivatives Tutorial

Find functional derivatives, which are the derivative of a functional with respect to a function.

Related Information

Featured Examples