General Mathematical Identities for Analytic Functions: Differentiation

General Identities

General Mathematical Identities for Analytic Functions

General Identities

Differentiation

Low-order differentiation

Derivatives of the first order

This limit defines the derivative of a function at the point , if it exists.

This formula reflects the property that a constant factor can be pulled out of the differentiation.

This formula reflects the property that the derivative of a sum (and difference) is equal to the sum (and difference) of the derivatives.

The product rule for differentiation shows that the derivative of a product is equal to the derivative of the first function multiplied by the second function plus the derivative of the second function multiplied by the first function.

The product rule for differentiation shows how to evaluate the derivative of the product of three functions.

The product rule for differentiation shows how to evaluate the derivative of the product of functions.

The quotient rule for differentiation shows that the derivative of the ratio is equal to the derivative of the numerator multiplied by the denominator minus the derivative of the denominator multiplied by the numerator, divided by the square of the denominator.

The quotient rule for differentiation has been generalized to the case when the numerator is the product of two functions.

The quotient rule for differentiation is generalized to the case when the denominator is the product of two functions.

This formula shows that the derivative of the sum is equal to the sum of the derivatives. For an infinite sum it is true under some restrictions on , which ensure the convergence of the series.

This formula shows that the derivative of a power series is equal to the corresponding sum of the derivatives. It is true inside the corresponding circle of convergence with radius .

This chain rule for differentiation shows that the derivative of composition is equal to the derivative of the outer function in the point , multiplied by the derivative of the inner function .

This chain rule for partial differentiation generalizes the previous chain rule for differentiation in the case of a function with two variables .

This chain rule for partial differentiation generalizes the chain rule for differentiation in the case of a function with several variables .

This formula shows that the derivative of the inverse function is equal to the reciprocal of the derivative of the direct function in the point .

This formula shows that the composition of the first derivatives is equal to the derivative of the second order.

This formula shows that the derivative of an indefinite integral produces the original function (the derivative is the inverse operation to the indefinite integration).

This formula shows that the derivative of a definite integral with respect to the upper limit produces the original function.

This formula shows that the derivative of a definite integral with respect to the low limit gives the original function with a negative sign.

This formula shows that the order of differentiation and definite integration can be changed if the limits of the integral do not depend on the variable of differentiation.

This formula reflects the general rule of differentiating an integral when its limits and its integrand depend on the variable of differentiation.

Derivatives of the second order

This limit defines the second derivative of a function at the point , if it exists.

This formula shows how to evaluate the second derivative of a general composition .

Symbolic differentiation

Definition

This limit defines the -order derivative of a function at the point , if it exists.

Converting to finite differences and back

Products

This rule is called the binomial differentiation rule for the -order derivative.

This rule is called the multinomial differentiation rule for the -order derivative.

Ratios

This formula shows how to evaluate an -order derivative for the quotient of two functions.

This formula shows how to evaluate an -order derivative for the quotient of two functions.

This formula shows how to evaluate an -order derivative for the quotient of two functions.

Power

This formula shows how to evaluate an -order derivative of the power .

Positive integer powers

This formula shows how to evaluate an -order derivative of the square .

This formula shows how to evaluate an -order derivative of the cube .

This formula shows how to evaluate an -order derivative of the fourth power .

This formula shows how to evaluate an -order derivative of the general integer power .

Negative integer powers

This formula shows how to evaluate an -order derivative of the reciprocal .

This formula shows how to evaluate an -order derivative of the reciprocal .

Log from function

This formula shows how to evaluate an -order derivative of the composition with logarithm .

Exp from function

This formula shows how to evaluate an -order derivative of the composition with exponential function .

This formula shows how to evaluate an -order derivative of the composition with exponential function .

This formula shows how to evaluate an -order derivative of the composition with exponential function .

Function from power

This formula shows how to evaluate an -order derivative of the composition with power function .

This formula shows how to evaluate an -order derivative of the composition .

This formula shows how to evaluate the derivative of the -order of the composition .

This formula shows how to evaluate an -order derivative of the composition .

Function from exponent

This formula shows how to evaluate an -order derivative of the composition .

Function from trigonometric functions

This formula shows how to evaluate an -order derivative of the composition .

Function from hyperbolic functions

This formula shows how to evaluate an -order derivative of the composition .

General compositions

This formula shows how to evaluate an -order derivative of the general composition .

This formula is called Faá di Bruno's formula.

This formula, Faá di Bruno's relation, shows how to evaluate an -order derivative of the general composition .

This formula shows how to evaluate an -order derivative of the general composition .

General power exponential compositions

This formula shows how to evaluate an -order derivative of the general power exponential composition .

This formula shows how to evaluate an -order derivative of the general power exponential composition .

Inverse function

This formula shows how to evaluate an -order derivative of the inverse function .

Repeated derivatives

This formula shows how to apply times the operation to the function .

Fractional integro-differentiation

The fractional integro‐derivative of the function with respect to is defined by the preceding formula, where the integration in Mathematica should be performed with the option GenerateConditions->False: Integrate[f[t](z-t)α+n-1Gamma[α+n],{t,0,z},GenerateConditionsFalse. This definition supports the Riemann‐Liouville‐Hadamard fractional left‐sided integro‐differentiation at the point 0.

This formula for the fractional integro‐derivative represents the fractional integral of the function with respect to . This integral is called the Abel integral.

This formula for the fractional integro‐derivative actually represents the fractional derivative of the function with respect to . This derivative includes the composition of the corresponding usual derivative of order and an Abel integral.

This formula shows how to evaluate the fractional integro‐derivative of the analytical function near the point .

This formula shows how to evaluate the fractional integro‐derivative of a function having Laurent series expansion, multiplied on near the point .

This formula shows that for the evaluation of the fractional integro‐derivative of the analytical function near the point , you need to re-expand this function in a series near the point and then evaluate the corresponding integro‐derivative.