How to Calculate Derivatives and Integrals on TI-84 — Complete Guide (2026)

The TI-84 handles two core calculus operations: numerical derivatives using nDeriv and definite integrals using fnInt. Both functions live in the MATH menu and work on any typed or graphed function. This guide covers every method — evaluating derivatives and integrals on the home screen, graphing the derivative of a function, reading dy/dx directly from a plotted curve, and shading the area under a curve — with worked examples for each. It also explains what "numerical" means for these functions and where the limitations are, so you use them correctly on AP Calculus exams and college assignments.

Numerical vs Symbolic — What TI-84 Actually Does

Before diving into syntax, one point worth clarifying upfront: the TI-84 is a numerical calculator, not a computer algebra system (CAS). It cannot produce symbolic results like "f'(x) = 2x + 3." What it can do is evaluate any function at specific numbers with high precision.

For AP Calculus AB and BC, these two functions handle the vast majority of calculator-required problems. If you need a symbolic derivative expression for an entire function, you would need a CAS calculator — but most standardized exams specifically allow only TI-84 class devices and test numerical skills.

Using nDeriv on the Home Screen

nDeriv takes three arguments: the function expression, the variable (always X on the home screen), and the x-value at which to evaluate the derivative.

Step-by-Step: Derivative at a Point

• 1
  Press MATH The MATH menu opens. You'll see a numbered list of functions.
• 2
  Press 8: nDeriv( The function name pastes to the home screen with an open parenthesis.
• 3
  Type the function, a comma, X, a comma, and the x-value Example for f(x) = x³ at x = 2: type nDeriv(X^3, X, 2)
• 4
  Press ) then ENTER The result appears: 12 (since f'(x) = 3x², and 3(2²) = 12).

Common nDeriv Examples

Optional Fourth Argument: Tolerance (ε)

nDeriv accepts a fourth optional argument that controls the step size used in the symmetric difference approximation. The default is 0.001 and is almost never changed. Syntax: nDeriv(f(x), X, value, ε). A smaller ε generally improves accuracy but can cause issues at discontinuities or sharp corners.

Using fnInt on the Home Screen

fnInt computes the definite integral of a function between a lower and upper bound. It returns the net signed area — regions below the x-axis contribute negatively.

Step-by-Step: Definite Integral

• 1
  Press MATH → 9: fnInt( The function pastes to the home screen.
• 2
  Type the function, variable X, lower bound, upper bound Example for ∫₀² x² dx: type fnInt(X^2, X, 0, 2)
• 3
  Press ) then ENTER Result: 2.6667 (exact answer is 8/3 ≈ 2.6667) ✓

Common fnInt Examples

Finding dy/dx Directly from a Graph

When a function is already plotted in Y₁, you can read the derivative value at any point on the curve without typing nDeriv manually. The CALC menu's dy/dx option does this interactively.

• 1
  Graph your function in Y₁ Press Y=, enter your function, press GRAPH. For a refresher on graphing steps, see the TI-84 graphing guide.
• 2
  Press 2nd → TRACE (CALC) The CALCULATE menu opens with options for value, zero, minimum, maximum, intersect, dy/dx, and ∫f(x)dx.
• 3
  Press 6: dy/dx The graph reappears with a cursor on the curve.
• 4
  Type a specific x-value and press ENTER, or use arrow keys to move the cursor The slope (derivative) at the cursor's current position appears at the bottom of the screen labeled dy/dx =.

Calculating Area Under a Curve on a Graph

The CALC menu also lets you select integration bounds visually on the graph and instantly see the shaded area with the numerical result — useful for problems where you need to illustrate the integral, not just compute it.

• 1
  Graph your function in Y₁ and set a clear window Ensure the region you want to integrate is visible on screen. Use ZOOM → ZStandard if needed.
• 2
  Press 2nd → TRACE (CALC) → 7: ∫f(x)dx The graph screen reappears.
• 3
  "Lower Limit?" — move the cursor or type the lower x-value, press ENTER A left boundary marker appears on the curve.
• 4
  "Upper Limit?" — move the cursor or type the upper x-value, press ENTER The region between the curve and x-axis is shaded, and the numerical result appears at the bottom labeled ∫f(x)dx =.

Graphing the Derivative Function

nDeriv can be embedded inside a Y= equation to plot the derivative as a full curve — not just at one point. This is useful for visually analyzing where a function is increasing, decreasing, or has critical points.

• 1
  Enter your original function in Y₁ Example: Y₁ = X³ − 3X
• 2
  Move to Y₂ and enter: nDeriv(Y₁, X, X) To reference Y₁: press VARS → 1: Function → 1: Y₁. This tells the calculator to differentiate Y₁ with respect to X, evaluated at each X pixel as it draws.
• 3
  Press GRAPH Both the original function and its derivative curve appear simultaneously. The derivative curve crosses zero exactly where the original function has local maxima or minima.

Worked Examples — AP Calculus Style

Example 1 — Slope of a Tangent Line

Problem: Find the slope of f(x) = 2x³ − 5x + 1 at x = 2.

Verification: f'(x) = 6x² − 5, so f'(2) = 6(4) − 5 = 19 ✓

Example 2 — Area Between Two Curves

Problem: Find the area between f(x) = x² and g(x) = x from x = 0 to x = 1.

The area is ∫₀¹ (x − x²) dx since g(x) ≥ f(x) on [0,1].

Exact answer: [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6 ≈ 0.1667 ✓

Example 3 — Rate of Change from a Graphed Function

Problem: Y₁ = sin(x) is graphed. Find the instantaneous rate of change at x = π/4.

• Press 2nd → TRACE → 6: dy/dx
• Type π/4 and press ENTER
• Result: dy/dx ≈ 0.7071

Exact: cos(π/4) = √2/2 ≈ 0.7071 ✓

nDeriv and fnInt Accuracy — What to Know

Both functions return approximations. For most well-behaved functions encountered in calculus courses, the results are accurate to 8–10 significant digits — more than sufficient for any exam. But there are specific situations where the approximations break down:

Common Mistakes and How to Fix Them

Related TI-84 Guides

Calculus work on the TI-84 often connects directly to graphing and algebra skills. These guides cover the techniques you'll use alongside derivatives and integrals:

Frequently Asked Questions

Putting It All Together

The TI-84's calculus toolkit is built around four functions you can access from two menus: nDeriv and fnInt on the home screen via MATH, and dy/dx and ∫f(x)dx on the graph via 2nd → TRACE. Together they cover every numerically-solvable calculus problem you'll encounter in AP Calculus AB, AP Calculus BC, and most college calculus courses.

The most important thing to keep in mind is that these functions produce numerical approximations — accurate to many decimal places for well-behaved functions, but not symbolic expressions. Use them to verify analytic work and evaluate definite integrals where exact antiderivatives are difficult. For problems requiring exact symbolic derivatives or antiderivatives, work through the algebra by hand and use the calculator to confirm the numerical result.

If your calculator starts behaving unexpectedly during calculus work — wrong trig results, strange mode errors — a quick Restore Defaults reset will fix most mode-related issues without erasing your programs. For a complete library of TI-84 function guides, visit the Guides & Tutorials section.