If you are studying for a course, include the allowed method. A correct answer obtained with a rule your exercise asks you to derive may leave the actual learning task unfinished.
Keep the point and the interval separate
Consider this original practice function: f(x) = x² + 3x. Find the derivative at x = 2 using the limit definition. This worked example was authored for the guide; it is not a ChatGPT transcript or a measured learning result.
First, f(2) = 10. For a nonzero increment h:
f(2 + h) = (2 + h)² + 3(2 + h)
= 10 + 7h + h²
[f(2 + h) - f(2)] / h = (7h + h²) / h = 7 + h, h ≠ 0
As h approaches 0, 7 + h approaches 7.
Therefore f'(2) = 7.
We cancel h while h is nonzero, then take the limit. Substituting h = 0 into the original quotient produces 0/0, which is undefined. A limit concerns the values approached nearby; it does not make that quotient defined at zero.
OpenStax's derivative definition, checked October 24, 2026, develops the derivative from secant slopes approaching a tangent slope. Here, that relationship explains the purpose of the limit calculation.
Use a numerical check without replacing the argument
| Nonzero h | Difference quotient 7 + h |
|---|---|
| 0.1 | 7.1 |
| 0.01 | 7.01 |
| -0.01 | 6.99 |
The values are consistent with a limit of 7. A few decimals alone do not prove a limit exists for an arbitrary function. For this polynomial, the exact expression 7 + h supplies the argument.
By contrast, the interval from x = 2 to x = 3 has average slope (18 - 10) / (3 - 2) = 8. That is a valid interval calculation answering a different question. A response reporting 8 as f'(2) has used a finite interval where the exercise asks for a limiting slope at a point.
Ask about the first step you cannot justify
OpenAI's prompting guidance, checked October 24, 2026, supports supplying context and specifying the desired result. For calculus practice, name the mathematical step you want explained.
Help me understand the derivative at a point using the limit definition.
Function: f(x) = x² + 3x. Point: x = 2.
My attempted work: [PASTE MY ACTUAL STEPS]
Course method: expand the difference quotient before taking the limit.
Check my first unjustified step. Explain what each quotient measures
and why h must be nonzero before cancellation. Ask me to expand f(2+h)
and wait for my attempt before showing the next step.
After we finish, compare the derivative with the average slope from 2 to 3.
Use two small nonzero increments as checks, clearly labeled estimates.
Do not treat a finite table as proof or substitute zero into a denominator.
Do not invent my progress, a grade or a claim that I mastered the topic.
Test a new function with the worked answer hidden
Try g(x) = x² + x at x = 3. Expanding the quotient gives 7 + h, so the derivative there is also 7. The matching answer does not mean the functions are identical; you have compared their slopes at two specified points.
Explain the expansion, the nonzero-h cancellation and the limiting step on paper. If any line still depends on copying the chat, use guided study practice to ask about that line. Keep the course's notation and your own derivation with the answer, and take the specific disputed step to your instructor when the explanation remains unclear.