D

D[f,x]

gives the partial derivative .

D[f,{x,n}]

gives the multiple derivative .

D[f,x,y,]

gives the partial derivative .

D[f,{x,n},{y,m},]

gives the multiple partial derivative .

D[f,{{x1,x2,}}]

for a scalar f gives the vector derivative .

D[f,{array}]

gives an array derivative.

Details and Options

  • D is also known as derivative for univariate functions.
  • By using the character , entered as pd or \[PartialD], with subscripts, derivatives can be entered as follows:
  • D[f,x]xf
    D[f,{x,n}]{x,n}f
    D[f,x,y]x,yf
    D[f,{{x,y}}]{{x,y}}f
  • The comma can be made invisible by using the character \[InvisibleComma] or ,.
  • The partial derivative D[f[x],x] is defined as , and higher derivatives D[f[x,y],x,y] are defined recursively as etc.
  • The order of derivatives n and m can be symbolic and they are assumed to be positive integers.
  • The derivative D[f[x],{x,n}] for a symbolic f is represented as Derivative[n][f][x].
  • For some functions f, Derivative[n][f][x] may not be known, but can be approximated by applying N. »
  • New derivative rules can be added by adding values to Derivative[n][f][x]. »
  • For lists, D[{f1,f2,},x] is equivalent to {D[f1,x],D[f2,x],} recursively. »
  • D[f,{array}] effectively threads D over each element of array.
  • D[f,{array,n}] is equivalent to D[f,{array},{array},], where {array} is repeated n times.
  • D[f,{array1},{array2},] is normally equivalent to First[Outer[D,{f},array1,array2,]]. »
  • Common array derivatives include:
  • D[f,{{x1,x2,}}]gradient{D[f,x1],D[f,x2],}
    D[f,{{x1,x2,},2}]Hessian{{D[f,x1,x1],D[f,x1,x2],},{D[f,x2,x1],D[f,x2,x2],},}
    D[{f1,f2,},{{x1,x2,}}]Jacobian{{D[f1,x1],D[f1,x2],},
    {D[f2,x1],D[f2,x2],},}
  • If f is a scalar and x={x1,}, then the multivariate Taylor series at x0={x01,} is given by:
  • ,
  • where fi=D[f,{x,i}]/.{x1x01,} is an array with tensor rank . »
  • If f and x are both arrays, then D[f,{x}] effectively threads first over each element of f, and then over each element of x. The result is an array with dimensions Join[Dimensions[f],Dimensions[x]]. »
  • D can formally differentiate operators such as integrals and sums, taking into account scoped variables as well as the structure of the particular operator.
  • Examples of operator derivatives include:
  • is not scoped by the integral
    is scoped by the integral
    is not scoped by the integral transform
    is scoped by by the integral transform
  • All expressions that do not explicitly depend on the variables given are taken to have zero partial derivative.
  • The setting NonConstants{u1,} specifies that ui depends on all variables x, y, etc. and does not have zero partial derivative. »

Examples

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Basic Examples  (7)

Derivative with respect to x:

Fourth derivative with respect to x:

Derivative of order n with respect to x:

Derivative with respect to x and y:

Derivative involving a symbolic function f:

Evaluate derivatives numerically:

Enter using pd, and subscripts using :

Scope  (82)

Basic Uses  (12)

Derivative of an expression with respect to x:

Second derivative:

Derivative of an expression at a point:

Derivative of a function at a general point x:

This can also be achieved using fluxion notation:

Derivative at the point x==5:

This can be found more easily using fluxion notation:

The third derivative at the point x==-1:

Derivative involving symbolic functions:

Partial derivatives of an expression with respect to x and y:

The mixed partial derivative :

The mixed partial derivative :

Differentiate with respect to a compound expression:

Differentiate with respect to different compound expressions:

Derivative of a vector expression:

Matrix expression:

Derivative of a nested list:

Vector derivative of an expression, also known as the gradient:

Second vector derivative, also known as the Hessian:

Matrix derivative:

Create a table of basic derivatives:

Symbolic Functions  (9)

Derivative of a symbolic function:

Substitute in a pure function for f to get a particular result:

Derivative of a sum:

Product:

Quotient:

The chain rule for composite functions:

Product rule for three functions:

State the rule using an Inactive derivative:

Partial derivative of a symbolic function:

Substitute in for f a pure function in two variables:

Derivative of a pure function with respect to non-argument parameters:

The result is the function that at point x gives the derivative of with respect to a:

Local variables are independent from the differentiation variable:

Derivative of a symbolic function at a point:

The same, using prime notation:

Derivative of an inverse function:

Product rule for derivatives of order n:

Chain rule:

Elementary Functions  (6)

Polynomial and rational functions:

Algebraic functions:

Trigonometric and inverse trigonometric functions:

Exponential and logarithmic functions:

Hyperbolic functions:

Create a function that turns a list of expressions into a nicely formatted table of derivatives:

Create a table of trigonometric derivatives:

Create a table of hyperbolic derivatives:

Special Functions  (8)

The logarithmic derivative of Gamma is the PolyGamma function:

Derivatives of Airy functions are given in terms of AiryAiPrime and AiryBiPrime:

The derivative of Zeta has a closed-form expression at the origin:

Special functions with elementary derivatives:

Special functions with derivatives expressed in terms of the same functions:

Derivative of JacobiSN:

Derivative of JacobiCD:

Derivative of LogIntegral:

Derivative of ExpIntegralEi:

Derivative of order n for SinIntegral:

Create a table of special function derivatives:

Piecewise and Generalized Functions  (8)

Derivative of a piecewise function:

Derivative of a ConditionalExpression:

Convert a symbolic function into a piecewise function over the reals to differentiate it:

Compute the piecewise derivative over a finite range:

Classical derivatives of pointwise-defined engineering functions:

Distributional derivatives of generalized functions:

Derivative of RealAbs:

RealSign:

Their counterparts on the complex plane are nowhere differentiable:

Derivative of Floor:

Ceiling:

Derivatives of functions defined procedurally:

Implicitly Defined Functions  (3)

D threads over Equal to compute the derivatives of implicit functions:

Compute a partial derivative for an implicit function of two variables:

Find partials for implicit functions defined by a system of equations:

Vector-Valued Functions  (5)

Derivative of a list:

Second derivative:

The first derivative to a vector-valued function at a general value t:

Computing the same using prime notation:

The third derivative at t==0:

Computing the same using prime notation:

The derivative of a matrix:

The fourth derivative:

The derivative of a vector-valued function stored as a SparseArray:

Convert the result to a normal array:

The derivative of matrix represented as a SymmetrizedArray object:

Convert the result to a normal matrix:

Vector Argument Functions  (6)

Gradient of a scalar function:

Hessian matrix:

Jacobian of a vector-valued function:

Second-order derivative tensor:

Compute the derivative of the determinant with respect to the original matrix:

The gradient of a vector-valued function stored as a SparseArray:

The result is another SparseArray, containing only the nonzero entries:

Convert the result to a normal matrix:

Hessian computed as a SparseArray:

The gradient can also be computed as a SparseArray, but in this case it is effectively dense:

Jacobian computed as a SparseArray:

Integrals and Integral Transforms  (6)

Differentiate unevaluated integrals:

Fourier transforms:

Laplace transforms:

Convolutions:

Differentiate the Inactive form of an integral to get the fundamental theorem of calculus:

A more general form of the fundamental theorem:

Differentiate an inactive FourierTransform:

Verify the formal result:

Sums and Summation Transforms  (5)

Differentiate an unevaluated sum:

Differentiation with respect to the dummy variable gives zero:

Z transforms and generating functions:

Discrete convolution:

Differentiate the Inactive form of a sum:

ZTransform:

Differentiate an inactive GeneratingFunction:

Verify the formal result:

Indexed Differentiation  (9)

Differentiate with respect to an indexed variable, introducing KroneckerDelta factors:

Use Inactive to prevent expansion of the sum:

Summation indices will be renamed if needed, to avoid name ambiguities:

Differentiate an inactive table with respect to an indexed variable:

Activate the result to get the explicit vector result:

Differentiate an inactive table twice with respect to an indexed variable:

In this case only the j^(th) entry is nonzero:

Use any notation for indexed variables in sums and tables:

Differentiate with respect to a symbolic table of indexed variables:

Activating the result gives the explicit gradient:

Differentiate twice with respect to a symbolic table of indexed variables, introducing a dummy index:

Replace symbolic variables with explicit values:

Use symbolic vector differentiation of another symbolic vector:

Vector differentiation of a vector with respect to itself gives the identity matrix:

Functions Defined by Derivatives  (5)

Define the derivative with prime notation:

This rule is used to evaluate the derivative:

Define the derivative at a point:

Define the second derivative:

Prescribe values and derivatives of f and g:

Find the derivative of the composition at x=3:

Define a partial derivative with Derivative:

Options  (1)

NonConstants  (1)

Differentiate with y considered as depending on x:

Solve for the derivative of y to effect implicit differentiation:

Applications  (41)

Geometry of the Derivative  (5)

The derivative gives the slope of the tangent line at a point:

For small displacements h from the base point π, the tangent line gives an excellent approximate of f:

The tangent and f are visually indistinguishable from each other over a small, and only a small, plot range:

The derivative gives the limit of the slope of the secant line connecting {x,f[x]} to {x+h,f[x+h]}:

Visualize the process for the point {1,f[1]}:

Find an equation for the tangent line to a function:

General equation for the tangent line at x=a:

Tangent line at x=4:

Find an equation for the normal line to a function:

General equation for the normal line at x=a:

Normal line at x=1:

Find equations for the tangent lines to a function that pass through a point:

General equation for the tangent line at x=a:

Find the tangents to f[x] that pass through (0,-4):

Characterization of Functions  (5)

Find the turning points on a plane curve:

Find the critical points of a function:

By the second derivative test, these are all maxima or minima:

Visualize the critical points:

Find all values of c that satisfy the Mean Value theorem on an interval:

Define the secant line from a to b:

Define the tangent lines associated with the two values of c:

Visualize the two tangent lines parallel to the secant line along with the original function:

Use the first derivative to characterize a function:

Find the critical points of a function:

Find where the function is increasing:

Find where the function is decreasing:

Visualize the results:

Use the second derivative to characterize a function:

Find the inflection points of a function:

Find where the function has positive concavity:

Find where the function has negative concavity:

Visualize the results:

Relation to Integration  (2)

Perform the change of variable t=x^2 in an integral:

Verify the results of symbolic integration:

Multivariate and Vector Calculus  (6)

Find the critical points of a function of two variables:

Compute the signs of and the determinant of the second partial derivatives:

By the second derivative test, the first two pointsred and blue in the plotare minima and the thirdgreen in the plotis a saddle point:

Find the curvature of a circular helix with radius r and pitch c:

Obtain the same result using ArcCurvature:

Compute a univariate Taylor series by hand:

Compute a multivariate Taylor series by hand:

Write a function to automate the process:

Recompute the above using the new function:

The gradient vector can be computed by finding the partial derivatives of a function:

Find the gradient vector of the function :

Visualize the direction of the gradient vector using a unit vector representation:

The curl of a vector field on the plane can be computed by subtracting the derivatives of its components:

Find the curl of the vector field :

Visualize the 2D curl as the net "rotation" of the vector field at a point, with red and green representing clockwise and counterclockwise curl, respectively, and radius proportional to the magnitude of rotation:

The divergence of a vector field can be computed by summing the derivatives of its components:

Find the divergence of a 2D vector field:

Visualize 2D divergence as the net "flow" of the vector field at a point, with red and green representing outflow and inflow, respectively, and radius proportional to the magnitude of the flow:

Differential Equations  (6)

Construct the differential equation satisfied by an implicit function y[x]:

Use D to specify ordinary and partial differential equations:

These can be solved using DSolve:

Define a wave equation in two spatial variables:

Define initial values for the function and its first time derivative:

Specify boundary conditions:

Solve the system using DSolve:

Extract a few terms from the Inactive sum:

The two-dimensional wave executes periodic motion in the vertical direction:

Specify a Laplacian operator using D:

Specify homogeneous Dirichlet boundary conditions:

Find the 4 smallest eigenvalues and eigenfunctions of the operator in a unit disk:

Visualize the eigenfunctions:

Specify an integro-differential equation using D:

Obtain the general solution:

Specify an initial condition to obtain a particular solution:

Plot the solutions for different values of a:

Find a second-degree polynomial solution to the differential equation:

Rates of Change  (5)

The height of a projectile at time t is given by:

Compute the velocity at t:

Compute the acceleration at t:

Find when the projectile reaches its maximum height:

Find the maximum height of the projectile:

The area of a circle as a function of time is given by:

Compute the rate of change of area:

Find the rate of change of area at a radius of 10 m if the radius increases at a rate of 5m/s:

The position of a particle is given by:

Compute the velocity, acceleration, jerk, snap (jounce), crackle and pop of the particle:

The total resistance in a circuit of two resistors connected in parallel is given by:

Calculate RT for the given values of R1 and R2:

Find the rate of change of the total resistance:

Calculate the rate of change of the total resistance with the given values:

Volume of a cube in terms of side length l is given by:

Surface area of a cube is given by:

Compute the rate of change of the volume of a cube with respect to surface area using the chain rule:

Solve for l in terms of surface area and substitute that into the result:

Implicit Functions  (3)

Find an equation for the tangent line to an implicit function at (1,):

Compute the slope at x=1:

Tangent line at x=1:

Visualize the tangent:

Find the points where the slope of an implicit function equals :

The relationship between functions is given by:

Find the derivative of z[t]:

Calculate z'[t] with the given values:

Optimization  (3)

Find the maximum area of a rectangular fence of 2000 ft., bordered on one side by a barn:

Compute the area in terms of width:

Find the maximum:

By the second derivative test, this value is a true maximum:

Alternately, compute the area in terms of length:

Visualize how the area changes as the length changes:

Find the shortest distance from a curve to the point (1,5):

Compute the distance in terms of y:

Find the minimum point:

By the second derivative test, this is a minimum:

Visualize how the distance changes with position:

Find the dimensions of a lidless, cylindrical can with the least material that can hold up to 2 L of water:

Compute the height in terms of the radius, using the volume constraint:

Compute the surface area in terms of the radius:

The radius corresponding to the minimum surface area:

By the second derivative test, this is a minimum:

Compute the radius, height and surface area of the minimum configuration:

Visualize how the dimensions vary with radius:

L'Hôpital's Rule  (3)

Find the limit of the ratio of two functions as x0:

Directly solving the limit leads to an indeterminate form of type :

L'Hôpital's rule can be used because in an interval around , both and are defined, and :

Indeed, and are continuous, and , so can be computed trivially:

Verify the result using Limit:

Visualize the two functions and their ratio:

Find the limit of the ratio of two functions as x:

Directly solving the limit leads to an indeterminate form of type :

L'Hôpital's rule can be used because for all , both and are defined, and :

However, using the first derivatives also leads to an indeterminate form:

The second derivatives are constant and obviously satisfy the conditions of L'Hôpital's rule:

Hence can be computed trivially:

Verify the result using Limit:

Find the limit of the product of two functions as x0:

Directly solving the limit leads to an indeterminate form of type 0×:

Note that is not defined:

However, exists and is positive for all , and it also exists and is negative for all :

As is clearly defined for all real , L'Hôpital's rule can be applied in the form:

The quotient in the right-hand limit gives a continuous expression whose limit is simple to compute:

Other Applications  (3)

Compute the coefficients of a power series:

Derive a closed form for by differentiating with respect to at :

Now integrate first and then differentiate with respect to at :

The final result:

Verify the result:

Derive a closed form for by differentiating w.r.t. at :

Compute and then differentiate:

The result:

Verify the result:

Properties & Relations  (21)

The derivative of a function is defined as a limit:

The Limit of DifferenceQuotient is the derivative D:

D is the inverse of Integrate:

The fundamental theorem of calculus:

Differentiation inside of Integrate:

D returns formal results in terms of Derivative:

D differentiates expressions with respect to a given variable:

Derivative is an operator and returns pure-function results:

The derivative of a function at a point may not be available in closed form:

An approximation to the derivative can be obtained using N:

D[f,{array1},] is essentially equivalent to First[Outer[D,{f},array1,]]:

If f and a are arrays, Dimensions[D[f,{a}]==Join[Dimensions[f],Dimensions[a]]:

D[f,{{x1,x2,,xn}}] is effectively equivalent to Grad[f,{x1,x2,,xn}]:

Div[{f1,f2,,fn},{x1,x2,,xn}] is the trace of the vector derivative of f:

More generally, Div[f,x] is the contraction of the last two dimensions of the vector derivative of f:

Curl[f,x] is times the HodgeDual of the vector derivative of f, where r is the rank of f:

For scalar f, Laplacian[f,{x1,x2,,xn}] is the trace of the second vector derivative of f:

More generally, Laplacian[f,x] is the contraction of the last two dimensions of the second vector derivative of f:

ArcCurvature can be defined in terms of D:

Systems of differential equations involving D can be solved with DSolve:

Use D to specify a heat equation with homogeneous Dirichlet boundary conditions:

The eigensystem for this differential system can be found with DEigensystem:

Eigenvalues:

Eigenfunctions:

D can be defined using DifferenceDelta:

D can be defined using DiscreteShift:

The right one-sided derivative is computed with a right-hand limit:

The left one-sided derivative is computed with a left-hand limit:

Note that this function is not differentiable at x==0:

D assumes that other variables are independent of the differentiation variable:

Dt assumes that other variables may depend on the differentiation variable:

By manually specifying all other variables as constant, Dt can yield the same result as D:

Possible Issues  (4)

Results may not immediately be given in the simplest possible form:

Functions given in different forms can yield the same derivatives:

D returns generic results that may not account for discontinuities, cusps or other special points:

Neither f nor g is differentiable at 0:

f is discontinuous, and g has a cusp:

If a function can be expanded into a Piecewise expression, D will provide more accurate results:

Cached values for D may miss changes in underlying definitions:

The issue can be resolved by clearing the system cache:

Interactive Examples  (2)

Find the tangent line to a function:

Visualize the secant converging to the tangent as for different base points :

Neat Examples  (2)

Compute the tangent and normal vectors of a 3D parametric function:

Create a table of n^(th) derivatives:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)
2004
 (5.1)
2008
 (7.0)
2017
 (11.1)