How to Find Derivatives on a TI-84 with nDeriv(

The TI-84 Plus and TI-84 Plus CE have no symbolic differentiation key, so they approximate the gradient of a curve rather than rearranging it. The tool for the job is MATH ▸ 8:nDeriv(, which estimates the derivative at one chosen point by sampling the function a hair either side of it and dividing the change in y by the change in x.

That approach is fast and accurate for the smooth curves you meet in calculus, and it is the standard way to check a derivative you worked out by hand. This page covers the syntax, the X,T,θ,n key, the optional tolerance argument, and the two situations where the numeric answer should not be trusted.

What nDeriv( is doing

nDeriv( does not know the derivative rule for your function. It evaluates the function at x + h and at x − h, where h is a very small step set by the tolerance, then works out (f(x + h) − f(x − h)) ÷ (2h). This is a symmetric difference, and for a smooth curve the small errors on each side largely cancel, which is why the answer is usually accurate to several decimal places.

It is still an approximation, so treat the result as a very good estimate rather than an exact value. If a question asks you to “show that” a derivative takes a certain value, the calculator confirms it; the algebra still needs to be written out.

Entering the command and the X key

Press MATH and scroll down to 8:nDeriv(, then press ENTER to paste it.

The syntax has three parts inside the brackets:

nDeriv(expression, X, value)

1. the expression you want to differentiate,
2. then a comma, the variable X, and another comma,
3. and finally the value of x at which you want the gradient.

For the variable, use the X,T,θ,n key, not a letter typed from the alphabet. The X,T,θ,n key inserts the variable the calculator recognises; a plain X produced with ALPHA is a different symbol and will not match. Getting a comma in the wrong place leads to ERR: SYNTAX, which the SYNTAX error page explains how to unpick.

The tolerance argument

There is an optional fourth part: nDeriv(expression, X, value, tolerance). The tolerance is the tiny step the calculator takes on each side of the point. Leave it out and the calculator uses a sensible default. Type a smaller number, such as 1E-6, and it takes a shorter step; type a larger one and it takes a longer step and the estimate becomes coarser.

Smaller is not automatically better. Once the step gets close to the limit of the calculator’s internal precision, rounding error starts to outweigh the gain, and the answer can get slightly worse rather than better. For schoolwork the default is fine; the tolerance argument is there for cases where a function changes very quickly near the point you are testing.

A worked example

Take f(x) = x³ − 2x and find the gradient at x = 2. Enter:

nDeriv(X³-2X,X,2)

and press ENTER.

By hand, f'(x) = 3x² − 2, and at x = 2 that is 3(4) − 2 = 10. The calculator agrees, which is exactly how this tool is meant to be used: the algebra gives the rule, and nDeriv( checks a single value against it.

Check the derivative on the graph

The second route is graphical and shows the geometry. Put the function in Y1, press GRAPH, then press 2nd TRACE to open the CALC menu and choose 6:dy/dx. Type the x-value and press ENTER: the calculator draws the tangent line and reports its slope at the bottom of the screen.

Seeing the tangent touch the curve at one point is a good sanity check. If the line cuts straight through the curve rather than grazing it, the x-value you entered is not where the derivative was wanted.

Where nDeriv( can mislead you

The method assumes the function is smooth near the point. Two situations break that assumption.

The first is a corner or a cusp. For f(x) = abs(X) at x = 0, the symmetric difference is (abs(h) − abs(−h)) ÷ 2h, which is zero, so nDeriv( reports a derivative of 0. That answer is wrong: abs(X) has no derivative at zero. A graph is the best warning here, because the corner is obvious on screen even though the number looks plausible.

The second is a point where the function, or the derivative, is undefined. Feeding such a point can produce ERR: DOMAIN, described on the DOMAIN error page, or a wildly large number that reflects the vertical slope rather than a real gradient.

When doing it by hand is faster

For a simple polynomial, differentiating by hand takes a few seconds and gives an exact expression you can then evaluate at any number of points. nDeriv( gives one number at one point, so it is the slower choice if a question asks for the derivative function rather than a single value.

Use the calculator to confirm one or two values, and reach for the algebra whenever a question wants the rule itself, a sign table, or the location of a turning point. The reverse tool, numeric integration, is covered on our definite integrals page, and the idea of a value a function approaches rather than reaches belongs with limits.

On the AP Calculus exams a graphing calculator such as the TI-84 Plus or Plus CE is permitted on the sections that allow one, but AP Calculus prohibits a calculator with a computer algebra system. The rules are updated from time to time, so confirm the current wording with the AP calculator policy and your teacher.