Property	Value
Hex Value	`$25`
Categories	Catalog > N Math > Math
Localizations	FR: `nDeriv(`

`nDeriv(`

Overview

When command is used in Classic mode, returns approximate numerical derivative of expression with respect to variable at value, with specific tolerance ε.
In MathPrint mode, numeric derivative template pastes and uses default tolerance ε.

Availability: Token available everywhere.

Syntax

nDeriv(expression,variable,value[,ε])

Arguments

Name	Type	Optional
expression	expression
variable
value
ε		Yes

Location

math, MATH, 8:nDeriv(

Description

nDeriv(f(var),var,value[,h]) computes an approximation to the value of the derivative of f(var) with respect to var at var=value. h is the step size used in the approximation of the derivative. The default value of h is 0.001.

nDeriv( only works for real numbers and expressions. nDeriv( can be used only once inside another instance of nDeriv(.

π→X
     3.141592654
nDeriv(sin(T),T,X)
     -.9999998333
nDeriv(sin(T),T,X,(abs(X)+E⁻6)E⁻6)
     -1.000000015
nDeriv(nDeriv(cos(U),U,T),T,X)
     .999999665

Advanced

If the default setting for h doesn't produce a good enough result, it can be difficult to choose a correct substitute. Although larger values of h naturally produce a larger margin of error, it's not always helpful to make h very small. If the difference between f(x+h) and f(x-h) is much smaller than the actual values of f(x+h) or f(x-h), then it will only be recorded in the last few significant digits, and therefore be imprecise.

A suitable compromise is to choose a tolerance h that's based on X. As suggested here, (abs(X)+]E⁻6)E⁻6 is a reasonably good value that often gives better results than the default.

Formula

The exact formula that the calculator uses to evaluate this function is:

(1) $\begin{align} \texttt{nDeriv}(f(t),t,x,h)=\frac{f(x+h)-f(x-h)}{2h} \end{align}$

This formula is known as the symmetric derivative, and using it generally increases the accuracy of the calculation. However, in a few instances it can give erroneous answers. One case where it gives false answers is with the function,

(2) $\begin{align} f(x) = \dfrac{1}{x^2} \bigg\vert_{x=0} \end{align}$

This derivative is undefined when calculated algebraically, but due to the method of calculation, the derivative given by nDeriv( is zero. These problems can be avoided by ensuring that a function's derivative is defined at the point of interest.

Error Conditions

ERR:DOMAIN is thrown if h is 0 (since this would yield division by 0 in the formula)
ERR:ILLEGAL NEST is thrown if nDeriv( commands are nested more than one level deep. Just having one nDeriv( command inside another is okay, though.

Source: parts of this page were written by the following TI|BD contributors: DarkerLine, Deoxal, GoVegan, kg583, Timothy Foster.

History

Calculator	OS Version	Description
TI-82	1.0	Added

Authors: Adrien Bertrand

nDeriv(