computes the ni^(th) derivative of data at level i.


computes the derivative at a Gaussian scale of standard deviation σ.


computes several derivatives der1, der2, .

Details and Options

  • DerivativeFilter is a linear filter that computes the derivatives of data based on a spline interpolation model. Regularization with a Gaussian kernel of standard deviation σ (defaults to 0) can be used to reduce susceptibility to noise.
  • The data can be any of the following:
  • listarbitrary-rank numerical array
    tseriestemporal data such as TimeSeries, TemporalData,
    imagearbitrary Image or Image3D object
    audioan Audio object
  • DerivativeFilter operates separately on each level of data.
  • DerivativeFilter[image,] uses the array coordinate system, where the first coordinate runs from the top to the bottom of image, and the second coordinate increases from left to right.
  • DerivativeFilter gives a result with the same dimensions as data.
  • DerivativeFilter can take the following options:
  • InterpolationOrderAutomaticinterpolation order up to 9
    Padding"Fixed"padding method
  • With Padding->{pad1,pad2,}, different padding schemes can be used for every dimension of data.
  • The derivative order has to be smaller than the specified interpolation order.


open allclose all

Basic Examples  (3)

A horizontal derivative of an image:

A regularized horizontal derivative of an image:

Derivative of a numeric list:

Scope  (13)

Data  (5)

First-order derivatives of a 2D array:

Obtain the first derivative of a TimeSeries object:

Filter an Audio signal:

Vertical derivative of a color image:

Vertical derivative of a 3D image:

Parameters  (8)

Zeroth derivative of a list:

First, second and third derivatives of a step sequence:

Vertical derivative of an image:

Horizontal derivative:

Second-order derivative in both dimensions:

Compute several derivatives of an image:

Vertical derivative of a 3D image:

Horizontal derivatives only:

Regularize the derivative using Gaussian smoothing:

Horizontal derivative at different Gaussian scales:

Options  (3)

InterpolationOrder  (1)

Filtering an array using different InterpolationOrder values:

Padding  (2)

Derivative filtering using different padding schemes:

First derivatives of a grayscale image using different padding schemes:

Use different padding schemes in each spatial direction:

Applications  (5)

Compute the image gradient:

Compute the Laplacian of an image at scale σ=6:

Ridge detection at scale σ=2:

T-junction filter:

Get borders from a colored map:

Properties & Relations  (4)

For larger values of , the results of GaussianFilter and DerivativeFilter converge:

DerivativeFilter and the corresponding derivatives of a spline interpolation return the same values:

Plot the result of the filter on top of the derivative of the interpolating function:

Derivative filtering of a binary image gives a grayscale image of a real type:

DerivativeFilter is a linear filter:

Wolfram Research (2010), DerivativeFilter, Wolfram Language function, (updated 2016).


Wolfram Research (2010), DerivativeFilter, Wolfram Language function, (updated 2016).


Wolfram Language. 2010. "DerivativeFilter." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016.


Wolfram Language. (2010). DerivativeFilter. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2021_derivativefilter, author="Wolfram Research", title="{DerivativeFilter}", year="2016", howpublished="\url{}", note=[Accessed: 27-January-2022 ]}


@online{reference.wolfram_2021_derivativefilter, organization={Wolfram Research}, title={DerivativeFilter}, year={2016}, url={}, note=[Accessed: 27-January-2022 ]}