What Is Increment Method?
The increment method is the first-principles way to differentiate. You add a small increment h to x, find how much y changes, and divide the change in y by h. That ratio is the slope of the secant line through two points on the curve.
Enter f(x), a point x and a value for h. The calculator returns the increment Δy = f(x + h) − f(x), the difference quotient, and a table where h shrinks from 1 down to 0.00001 so you can watch the quotient approach the derivative.
The table is a numerical check, not a proof. To prove a derivative by the increment method you still simplify the quotient algebraically and then let h go to zero, as shown in the worked example.
How to Use the Increment Method Calculator
- Type f(x) using x, numbers and operators.
- Enter the point x and an increment h.
- Read Δy and the quotient, then check the table to see the quotient settle as h gets smaller.
Increment Method Formula
Increment in y
Difference quotient
Derivative as a limit
Increment Method Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| f(x) | The function you are differentiating | |
| x | The point where the slope is measured | |
| h or Δx | The increment added to x | Must not be 0 |
| Δy | Change in y: f(x + h) − f(x) | |
| Δy / h | Difference quotient, the slope of the secant |
Increment Method Worked Example
With the example values loaded in the calculator above, the steps are:
- Δx = h = 0.1, so x + h = 2.1.
- f(2) = 10 and f(2.1) = 10.71.
- Δy = 10.71 − 10 = 0.71.
- Δy / Δx = 0.71 ÷ 0.1 = 7.1.
- Shrinking h toward 0 makes this quotient approach the derivative, as the table shows.
Result: Increment in y, Δy = f(x + h) − f(x) = 0.71; Difference quotient Δy / h = 7.1.
Worked example: f(x) = x² + 3x at x = 2
With h = 0.1: f(2) = 10 and f(2.1) = 10.71, so Δy = 0.71 and Δy / h = 7.1.
Algebraically, f(x + h) − f(x) = 2xh + h² + 3h, so the quotient is 2x + h + 3. At x = 2 that is 7 + h. As h shrinks, the quotient approaches 7, which is f′(2).
What the table shows and what it does not
The table is a numerical check. It shows the quotient approaching a value, but it is not a proof of the limit. For a proof, simplify the quotient algebraically before letting h go to 0.
Very small h can lose accuracy on a computer because f(x + h) and f(x) become nearly equal. That is why the tool stops at h = 0.00001.
Where Increment Method Is Used
Learning derivatives from first principles
Check each line of a textbook increment method problem with a number.
Estimating a rate of change
Get the average rate of change over a small interval when you only have the function.
Testing differentiability
Compare the quotient with h and −h to see whether a corner or jump breaks the limit.
Common Increment Method Mistakes to Avoid
- Use a negative h to see the backward difference quotient.
- If the quotient from h and −h disagree as they shrink, the function may not be differentiable at that point.
Increment Method Terms and Definitions
- Increment
- A small change in a variable, written Δx or h.
- Secant line
- The line through (x, f(x)) and (x + h, f(x + h)). Its slope is the difference quotient.
Increment Method FAQs
What is the increment method?
It is the first-principles way to find a derivative: add an increment h to x, form [f(x + h) − f(x)] ÷ h, simplify, then let h approach 0.
What is the difference between Δy and dy?
Δy is the actual change in f over the interval. dy = f′(x)·dx is the change along the tangent line. They are close when dx is small.
Can the calculator give a symbolic derivative?
No. It computes numbers: Δy, the quotient and a limit estimate. It does not simplify the expression symbolically.
Why does the quotient equal the slope of a secant line?
Δy ÷ Δx is rise over run between two points on the curve, which is the slope of the line joining them.
What does it mean if the limit does not exist?
The function has a corner, jump or vertical tangent there, so it has no derivative at that point.
Which functions can I enter?
Polynomials and expressions using sin, cos, tan, ln, log, sqrt, exp and abs, with ^ for powers.