Rational-power derivatives and valid domains
| English | 中文 | Pinyin |
|---|---|---|
| rational power | 有理数指数 | yǒu lǐ shù zhǐ shù |
A square-root curve has a finite height at zero but an unbounded slope nearby. Why must a derivative have its own domain?
- A square-root curve has a finite height at zero but an unbounded slope nearby. Why must a derivative have its own domain?
- This lesson studies rational power 有理数指数: An exponent that can be written as a fraction of integers, interpreted with a valid real root.
Choose the mathematical structure
- For a rational exponent p, d(x^p)/dx=p x^(p−1) wherever the real function is defined and differentiable. Multiply by constant coefficients; differentiate sums and differences term by term. A constant has derivative zero. Rewrite radicals and reciprocals as powers before subtracting one from the exponent. Positive x is valid for all rational powers; negative and zero inputs need separate checks.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines rational power?
An exponent that can be written as a fraction of integers, interpreted with a valid real root.
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 x>0, f=3√x−4/x+1/√x gives f′=(3/2)x^(−1/2)+4x^(−2)−(1/2)x^(−3/2). At x=4 this is 3/4+1/4−1/16=15/16. For q=x^(2/3)=(cube root x)², negative x is allowed; q′=(2/3)x^(−1/3) for x≠0, so q′(−8)=−1/3. At zero q has a cusp: the difference quotient |h|^(2/3)/h diverges with opposite signs. For r=x^(4/3), [r(h)−r(0)]/h=cube root h tends to zero, so r′(0)=0. Thus a fractional power does not automatically exclude zero from the derivative.
Rational-power derivatives and valid domains
For a rational exponent p, d(x^p)/dx=p x^(p−1) wherever the real function is defined and differentiable
Choose the derivative rule and preserve every coefficient, inner rate and input restriction.
For the worked f, find f′(4).
(3/2)/2+4/16−(1/2)/8=15/16.
Test a tempting shortcut
- The original root domain and derivative domain can differ. √x is defined at x=0, but 1/(2√x) has no finite value there; 1/√x is not defined there at all. A real negative base needs a reduced rational exponent with an odd denominator. For x<0, x^(2/3) means the square of its real cube root, not an invalid real logarithm calculation. Do not differentiate a reciprocal by keeping the same exponent or dropping the minus sign.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every rational-power function defined at zero has a finite derivative there. This claim is false. Explain which definition or assumption it violates.
For x^(2/3), find the derivative at x=−8.
The real cube root of −8 is −2; (2/3)/(−2)=−1/3.
Every rational-power function defined at zero has a finite derivative there.
The original root domain and derivative domain can differ. √x is defined at x=0, but 1/(2√x) has no finite value there; 1/√x is not defined there at all. A real negative base needs a reduced rational exponent with an odd denominator. For x<0, x^(2/3) means the square of its real cube root, not an invalid real logarithm calculation. Do not differentiate a reciprocal by keeping the same exponent or dropping the minus sign.
Interpret a new situation
- State the original domain first, rewrite each term, multiply its coefficient by the exponent and subtract one from that exponent. Return to radicals if requested. Check boundary inputs with the original difference quotient when the formal derivative is undefined or inconclusive. Keep exact fractional coefficients until the final evaluation.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For x^(4/3), find the derivative at x=0.
The original difference quotient is the real cube root of h and tends to zero.
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.
An exponent that can be written as a fraction of integers, interpreted with a valid real root. Choose the relationship, show the method, check its assumptions and interpret the result.