The nDeriv( Command

Command Summary

Calculates the approximate numerical derivative of a function, at a point.

Command Syntax

nDeriv(f(variable),variable,value[,h])

Press:

1. MATH to access the math menu.
2. 8 to select nDeriv(, or use arrows.

TI-83/84/+/SE

1 byte

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


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} \operatorname{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.

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