Compute a forward difference quotient with step h:

Backward difference quotient:

Symmetric difference quotient:

Compute the second difference quotient with step h:

Third difference quotient:

Partial difference quotient with steps r and s:

DifferenceQuotient threads over lists:

DifferenceQuotient of a constant is 0:

DifferenceQuotient of a polynomial function is a polynomial function:

Each successive difference quotient will lower the degree in x by one:

Rational functions:

Difference quotients of rational functions will stay as rational functions:

Trigonometric functions:

Exponential functions:

Polynomial exponentials:

Difference quotients of PolyGamma with an integer step are rational functions:

Similarly for HarmonicNumber and Zeta:

FactorialPower with step h has a simple difference quotient for a matching step h:

DifferenceQuotient of a multivariate polynomial function is a polynomial function:

Difference quotients of multivariate rational functions will stay as rational functions:

Higher-order quotients will tend to grow in size:

DifferenceQuotient of multivariate functions depending on a subset of the variables are 0:

DifferenceQuotient for a product of univariate functions:

This is equal to the product of the individual difference quotients:

Specify assumptions on the variable x and the step h to obtain simpler results:

Compute the derivative of a polynomial from first principles:

Compute the derivative using D:

Exponential function:

Trigonometric function:

Compute the second derivative for a power function:

Third derivative for a power tower:

Compute the partial derivative with respect to x for a function of two variables:

Partial derivative with respect to y:

Mixed partial derivative:

Approximate the derivative at a point using DifferenceQuotient:

Derivative at x=2.7:

Approximation given by DifferenceQuotient:

Approximate the derivative of a function using various difference quotients:

Exact derivative:

Use forward difference quotients to obtain an approximation:

Backward difference quotients:

Symmetric difference quotients:

Approximate the partial derivatives at a point using DifferenceQuotient:

Discretize a differential equation using forward differences:

Solve the differential equation using DSolveValue:

Solve the difference equation using RSolveValue:

Compare the exact and approximate solutions:

Discretize a differential equation using backward differences:

Solve the differential equation using DSolveValue:

Solve the difference equation using RSolveValue:

Compare the exact and approximate solutions:

Discretize a differential equation using symbolic differences:

Solve the differential equation using DSolveValue:

Solve the difference equation using RSolveValue:

Compare the exact and approximate solutions using forward and backward differences:

Obtain the exact solution as a limit of the approximate solution:

Richardson extrapolation is a method for sequence acceleration that can be used to improve the rate of convergence of a sequence a[h], which depends on a parameter h. Apply Richardson extrapolation to accelerate the convergence of DifferenceQuotient to the derivative of a function f[x] using the sequence a[x,h], which is defined by:

Set up a scheme for Richardson extrapolation:

Define the function:

Compute the derivative at a point:

Approximation given by DifferenceQuotient:

Richardson extrapolation improves the derivative approximation: