Product and quotient rules with retained exclusions
| English | 中文 | Pinyin |
|---|---|---|
| quotient rule/ˈkwəʊʃənt ruːl/ | 商法则 | shāng fǎ zé |
Both factors in a changing area can vary. Why does multiplying their separate derivatives miss most of the rate?
- Both factors in a changing area can vary. Why does multiplying their separate derivatives miss most of the rate?
- This lesson studies quotient rule 商法则: The rule for differentiating a ratio of two changing functions when the denominator is nonzero.
Choose the mathematical structure
- For differentiable u,v, (uv)′=u′v+uv′. For v≠0, (u/v)′=(u′v−uv′)/v². The product rule has two contributions: change in either factor while the other provides its current value. The quotient rule uses the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all over v².
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines quotient rule?
The rule for differentiating a ratio of two changing functions when the denominator is nonzero.
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 y=x²e^x, y′=2xe^x+x²e^x=e^x(x²+2x), so y′(1)=3e. For z=(x²+1)/(x−1), x≠1, z′=[2x(x−1)−(x²+1)]/(x−1)²=(x²−2x−1)/(x−1)²; at x=2 it is −1. Polynomial division gives z=x+1+2/(x−1), hence z′=1−2/(x−1)², an independent check. For w=(x²−1)/(x−1), x≠1, cancellation gives w=x+1 and w′=1 only on that original domain. The missing point at x=1 is not restored; its continuous extension would be a different function.
Product and quotient rules with retained exclusions
For differentiable u,v, (uv)′=u′v+uv′
Choose the derivative rule and preserve every coefficient, inner rate and input restriction.
For x²e^x, find its derivative at x=0.
e⁰(0²+2×0)=0.
Test a tempting shortcut
- Multiplying u′v′ is not the derivative of uv. Reversing the quotient numerator flips its sign. The squared denominator does not eliminate the original pole or hole. Simplifying before differentiating can help, but it must preserve exclusions. A stationary quotient point must also lie in the original domain.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Cancelling a factor from a quotient automatically restores every excluded input. This claim is false. Explain which definition or assumption it violates.
For the worked z, find z′(2).
(4−4−1)/(2−1)²=−1.
Cancelling a factor from a quotient automatically restores every excluded input.
Multiplying u′v′ is not the derivative of uv. Reversing the quotient numerator flips its sign. The squared denominator does not eliminate the original pole or hole. Simplifying before differentiating can help, but it must preserve exclusions. A stationary quotient point must also lie in the original domain.
Interpret a new situation
- Name u,v and write u′,v′ before substitution. Keep brackets around both quotient numerator terms until expansion is complete. Factor a product derivative only after collecting both contributions. Compare an expanded or divided form when available and substitute into the original domain before evaluating or solving y′=0.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For w=(x²−1)/(x−1), find w′(2).
Cancellation gives x+1 only for x≠1; its derivative at the allowed input 2 is 1.
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 rule for differentiating a ratio of two changing functions when the denominator is nonzero. Choose the relationship, show the method, check its assumptions and interpret the result.