Challenge enrichment: rational powers and derivative domains

Before differentiating the displayed expression, which rewrite is correct?

Try all five exponent numerators. Compare the old coefficient and exponent with the derivative pair

Which rule works at every setting?

Rational powers in the derivative catalogue

Methods Units 3–4 extends the polynomial power rule to rational powers. Rewrite roots and reciprocals as powers, differentiate at interior inputs where the real function is differentiable, and check zero separately

1. 1Rewrite roots and reciprocals Write roots as fractional powers and reciprocals as negative powers before applying the rule \(\displaystyle \sqrt{x}=x^{1/2},\quad \frac{1}{x^{2}}=x^{-2}\)
2. 2Multiply the coefficient and exponent For a term ax^n, the new coefficient is a multiplied by n. \(\displaystyle 5x^{3}\quad\Rightarrow\quad 5\times3x^{2}=15x^{2}\)
3. 3Reduce the exponent by one After multiplying the coefficient, replace n by n-1 in the power \(\displaystyle x^{n}\quad\Rightarrow\quad nx^{n-1}\)
4. 4Drop constant terms A constant has no change as x changes, so its derivative is zero \(\displaystyle \frac{d}{dx}\left(-7\right)=0\)

Given

\(\displaystyle a=4,\quad n=\frac32\)

For 4x^(3/2), find the derivative coefficient

Given

\(\displaystyle n=\frac32\)

Using the given exponent, find the derivative exponent

Which power represents the real cube root of x²?

Read the real domain from a reduced exponent

For x^(p/q), reduce p/q first. Even q needs x ≥ 0; odd q allows either sign. Negative exponents additionally require x ≠ 0. A denominator in the printed expression may impose another exclusion

1. 1Reduce the exponent before deciding the real domain For reduced p/q, odd q allows negative inputs; even q requires nonnegative inputs. A negative power or an original denominator also excludes zero

What is the real domain of 1/√x?

Given

\(\displaystyle f(x)=3\sqrt{x},\quad x=4\)

Evaluate the derivative of 3√x at x = 4.

Ordinary derivatives and endpoint derivatives

In this packet, ordinary derivative means a finite two-sided derivative on an open real neighbourhood. A natural domain endpoint is handled separately with a right derivative. Do not silently change this convention when reporting a derivative domain

1. 1State the derivative convention This packet calls a finite two-sided derivative on an open real neighbourhood the ordinary derivative. At a natural endpoint, a finite right derivative is reported separately
2. 2Check zero with a difference quotient For nonzero a and 0 < n < 1, the difference quotient of ax^n is unbounded near zero. For n > 1 it tends to zero through admissible inputs. Odd-root domains permit both sides; even-root domains permit only the right side

For √x at zero, what happens to the right difference quotient?

For x^(4/3) at zero, does the ordinary two-sided derivative exist?

A finite right derivative can coexist with an excluded endpoint

For f(x) = x^(3/2), the natural domain is [0,∞). The right difference quotient is √h and tends to zero. Under our ordinary two-sided convention, D(f′) = (0,∞), while the right derivative at zero is separately zero. The formula (3/2)√x being defined at zero does not supply a missing left neighbourhood

1. 1State the derivative convention This packet calls a finite two-sided derivative on an open real neighbourhood the ordinary derivative. At a natural endpoint, a finite right derivative is reported separately

Choose the correct report for x^(3/2) at its natural endpoint zero

Simplify while retaining restrictions

Divide the numerator terms by √x before differentiating. Since that root is in a denominator, retain x > 0 even if the simplified expression has a value at zero

1. 1Keep restrictions after cancellation Simplifying an expression does not define the original function at a cancelled denominator zero. Its derivative cannot exist where the original function is undefined

Given

\(\displaystyle \frac{6x^3+2x}{\sqrt{x}}\)

After simplifying the printed quotient, what is the highest original exponent?

Given

\(\displaystyle 6x^{5/2}\)

Find the derivative coefficient of the given term

For x(x^(4/3) + 2)/x, what is the domain of the original function and its ordinary derivative?

Use exact-friendly inputs

At x = −8, the real cube root is −2. Evaluate powers through this real root. A negative exponent takes a reciprocal; its numerator controls the remaining power

1. 1Rewrite roots and reciprocals Write roots as fractional powers and reciprocals as negative powers before applying the rule \(\displaystyle \sqrt{x}=x^{1/2},\quad \frac{1}{x^{2}}=x^{-2}\)

Given

\(\displaystyle f(x)=3x^{-2/3},\quad x=-8\)

Evaluate the derivative of 3x^(−2/3) at x = −8.