Derivatives, Integrals, and Limits on a TI-84

The TI-84 handles derivatives and integrals two ways — a typed function on the home screen, or a point-and-click tool on the graph screen — and it doesn’t handle limits at all, for a reason worth understanding rather than just accepting.

nDeriv( and fnInt( on the home screen

nDeriv(expression, variable, value) returns the numerical derivative of expression with respect to variable, evaluated at value. For d/dx(x³) at x = 2:

nDeriv(X^3, X, 2)

fnInt(expression, variable, lower, upper) returns the numerical definite integral over [lower, upper]:

fnInt(X^2, X, 0, 3)

Both live under MATH on the keyboard (not a separate calculus menu), and both are numerical approximations — fast, accurate enough for coursework, but not a symbolic result. nDeriv(X^2,X,3) gives you 6, the number, not 2X the expression.

dy/dx and ∫f(x)dx on the graph screen

The same two operations exist as point-and-click tools once a function is graphed: press 2ND + TRACE (the CALC menu) and pick 6: dy/dx or 7: ∫f(x)dx. dy/dx asks for a single x-value and reports the slope of the tangent line there; ∫f(x)dx asks for a left and right bound and reports the signed area between the curve and the x-axis, shading the region on screen while it does. This is the same menu that holds zero, minimum, maximum, and intersect — see How to Graph on a TI-84 for the rest of that menu.

Why there’s no limit( function

This isn’t an oversight — it follows from the same thing covered in Is a TI-84 a CAS Calculator?: a limit, properly, is a symbolic question (“what value does this expression approach”), and the TI-84 has no symbolic algebra engine to answer it with. nDeriv( and fnInt( get away with being purely numerical because a derivative-at-a-point and a definite integral both reduce to a single number. A limit doesn’t reduce the same way — the honest workaround is evaluating the expression at x-values that step closer and closer to the point in question and watching where the output trends, which is a manual process, not a built-in one.

Try it yourself

The TI-84 calculator here implements both paths: type nderiv( or fnint( directly into the Home tab (lowercase, same argument order — expression, variable, then value or bounds), or switch to Graph mode and use the dy/dx and ∫f(x)dx tools under the graph’s analysis menu, which shade the integral region the same way the physical device does.