Secants, limits and first-principles power derivatives
| English | Español |
|---|---|
| difference quotient/ˈdɪfrəns ˈkwəʊʃənt/ | cociente de diferencias |
A distance sensor gives two positions a short time apart. What happens to their average rate as the interval shrinks?
- A distance sensor gives two positions a short time apart. What happens to their average rate as the interval shrinks?
- This lesson studies difference quotient 差商: The change in function output divided by a nonzero change in input.
Choose the mathematical structure
- For h≠0, the secant gradient is [f(x+h)−f(x)]/h. The derivative f′(x) is its limit as h tends to zero, if the two-sided finite limit exists. It gives the tangent gradient and instantaneous rate. The variable h may approach zero from either sign; simplify before taking the limit.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines difference quotient?
The change in function output divided by a nonzero change in input.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For f(x)=x², [(x+h)²−x²]/h=2x+h, so f′(x)=2x. At x=1, h=1,0.5,−0.5 give secant gradients 3,2.5,1.5, tending to 2 from both sides. The tangent through (1,1) is y−1=2(x−1). For f(x)=x³, expanding gives 3x²+3xh+h², hence f′(x)=3x². A constant gives zero and f(x)=x gives one. For any positive integer n, the binomial expansion leaves nx^(n−1) plus terms containing h, so the limit is nx^(n−1). This argument alone does not prove the rule for rational powers.
Secants, limits and first-principles power derivatives
For h≠0, the secant gradient is [f(x+h)−f(x)]/h
Connect a derivative calculation to its limit or gradient sign interpretation.
Find the derivative of x² at x=1.
The simplified difference quotient tends to 2x; at x=1 this is 2.
Test a tempting shortcut
- Substituting h=0 into the unsimplified quotient gives 0/0, not a gradient. Cancelling h is valid for h≠0 before the limit. Continuity alone is insufficient: at x=0, |x| has right quotient 1 and left quotient −1, so no derivative there. The tangent is a local line, not an exact replacement for the entire curve.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A difference quotient can be evaluated by setting h=0 before simplification. This claim is false. Explain which definition or assumption it violates.
For x³, find f′(2).
The cubic quotient tends to 3x²; at x=2 this is 12.
A difference quotient can be evaluated by setting h=0 before simplification.
Substituting h=0 into the unsimplified quotient gives 0/0, not a gradient. Cancelling h is valid for h≠0 before the limit. Continuity alone is insufficient: at x=0, |x| has right quotient 1 and left quotient −1, so no derivative there. The tangent is a local line, not an exact replacement for the entire curve.
Interpret a new situation
- Write the difference quotient, expand, cancel only with h≠0 stated, and take the limit. Interpret derivative units as output units divided by input units. When a question says first principles, merely quoting nx^(n−1) does not establish the limit. At a corner check both approach directions.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For x² at x=1 with h=0.5, find the secant gradient.
2x+h at x=1,h=0.5 gives 2.5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · G. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The change in function output divided by a nonzero change in input. Choose the relationship, show the method, check its assumptions and interpret the result.