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:

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:

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:

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:

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:

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:

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:

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 :

Derivative of a symbolic vector-valued function with respect to a scalar argument:

Derivative of a symbolic matrix–valued function with respect to a scalar argument:

Derivative of a symbolic array–valued function with respect to a scalar argument:

Derivatives of scalar-valued functions with respect to symbolic vector arguments:

Real symbolic vector argument:

Derivatives of scalar-valued functions with respect to symbolic matrix arguments:

Real symbolic matrix argument:

Derivative of a scalar-valued function with respect to a symbolic array argument:

Derivatives of symbolic array–valued functions with respect to symbolic array arguments:

Derivative of a composition of symbolic array functions:

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:

Differentiate an unevaluated sum:

Differentiation with respect to the dummy variable gives zero:

Discrete convolution:

Differentiate the Inactive form of a sum:

ZTransform :

Differentiate an inactive GeneratingFunction :

Verify the formal result:

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:

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 :

Differentiate with y considered as depending on x:

Solve for the derivative of y to effect implicit differentiation:

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):

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:

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

Verify the results of symbolic integration:

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 points—red and blue in the plot—are minima and the third—green in the plot—is 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:

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:

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 R_T 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:

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:

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:

Find the limit of the ratio of two functions as x0:

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 x0:

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:

Approximate the variance for a perturbed vector:

Order zero approximation:

Compare with the exact value:

Order one approximation:

Compare with the exact value:

Order two approximation:

Since the second derivative does not depend on , the order two approximation equals the exact value:

Approximate the determinant of a perturbed matrix:

Order zero approximation:

Compare with the exact value:

Order one approximation:

Compare with the exact value:

Order two approximation:

Compare with the exact value:

Derive a least-squares solution for data given as a list of pairs :

Find the vector of vertical deviations for the data:

Define the sum of squares of the vertical deviations for the data:

Set up the least-squares equations:

Generate some data:

Solve the least-squares problem for this data:

Find GammaDistribution parameters that best fit the given data using the maximum likelihood method:

Maximize the log-likelihood function :

Compute the gradient of :

Find a zero of the gradient, with replaced by :

Visualize the result:

Compare with the result computed using EstimatedDistribution :

Find an optimality condition for a portfolio optimization problem with the expected return and standard deviation :

The goal is to maximize when the vector of asset weights satisfies Total [x]=1. The constraint can be used to represent where the unconstrained vector variable consists of the first coordinates of :

The maximum occurs at a critical point of :

Express the condition in terms of :

Compute the gradient of the log-likelihood function of the linear regression model represented by the equation , where are normally distributed random variables with mean zero and variance :

The log-likelihood function is given by:

Compute :

Express the result in terms of :

Compute :

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: