Rational-power derivatives and valid domains
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| rational power/ˈræʃənl ˈpaʊə/ | lũy thừa hữu tỷ |
Why must a derivative have its own domain?
- A square-root curve has a finite height at zero but an unbounded slope nearby.
- 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.
The method
- 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.
√x is defined at x=0. Why is its derivative not finite there?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Existence of the FUNCTION at a point does not give a finite derivative: the difference quotient for √x at 0 is 1/√h, which blows up. Domain of f and domain of f′ must be checked separately.
A worked case
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).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
(3/2)/2+4/16−(1/2)/8=15/16.
Where it goes wrong
- 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.
Not true: Every rational-power function defined at zero has a finite derivative there. x^(2/3) is defined at 0, but its derivative (2/3)x^(−1/3) is undefined there.
For x^(2/3), find the derivative at x=−8. Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
x^(2/3) is defined at 0, but its derivative (2/3)x^(−1/3) is undefined there.
Applying it
- 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.
For x^(4/3), find the derivative at x=0.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The original difference quotient is the real cube root of h and tends to zero.
Match each power rewrite to its derivative.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Rewrite radicals and reciprocals as powers first, then apply px^(p-1) with its coefficient; the exponent drops by exactly one.
In your course
- AQA A-level Mathematics (7357), specification section G.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Rational power: An exponent that can be written as a fraction of integers, interpreted with a valid real root.
- For a rational exponent p, d(x^p)/dx=p x^(p−1) wherever the real function is defined and differentiable.
- Watch for: the original root domain and derivative domain can differ.