Derive a Quadratic’s Instantaneous Slope
A difference quotient measures the secant slope over a nonzero input change.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: For f(x)=x², find f′(2) from the limit of [f(2+h)−f(2)]/h. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8.
- Independent check inputs: For f(x)=x², find f′(3) from the limit of [f(3+h)−f(3)]/h. Candidate answers: (A) 12 (B) 5 (C) 6 (D) 7.
Learn the idea
A derivative measures instantaneous rate through a limiting difference quotient. Differentiation rules summarize that process for suitable functions, while a tangent line uses the derivative slope and the function’s actual point to model local behavior. A difference quotient measures the secant slope over a nonzero input change. Expanding and canceling common terms reveals a simpler expression whose limit as the change tends to zero gives the derivative.
Here is the method on the supplied teaching case. Expand: [(2+h)²−4]/h=[4h+h²]/h=4+h for h≠0. Taking h→0 gives 4. The derivative is the limiting secant slope.
Do not substitute zero into the original quotient before simplifying, because it divides by the change. Cancel only for nonzero change, then take the limit; the derivative is not just a finite-step average.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: For f(x)=x², find f′(2) from the limit of [f(2+h)−f(2)]/h. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. Solution: Expand: [(2+h)²−4]/h=[4h+h²]/h=4+h for h≠0. Taking h→0 gives 4. The derivative is the limiting secant slope. The final answer is 4. Do not substitute zero into the original quotient before simplifying, because it divides by the change. Cancel only for nonzero change, then take the limit; the derivative is not just a finite-step average.
Check your reasoning
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Helpful tips
- A difference quotient measures the secant slope over a nonzero input change.
- Do not substitute zero into the original quotient before simplifying, because it divides by the change.
See the answer and explanation
6
Expand: [(3+h)²−9]/h=[6h+h²]/h=6+h for h≠0. Taking h→0 gives 6. The derivative is the limiting secant slope.
Take it further
Changed-input task: For f(x)=x², find f′(9) from the limit of [f(9+h)−f(9)]/h. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. Show the mathematical reason, not only the selected answer. Expected result: 18. Expand: [(9+h)²−81]/h=[18h+h²]/h=18+h for h≠0. Taking h→0 gives 18. The derivative is the limiting secant slope.
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