1 of 8 0-day streak

Challenge enrichment: rational powers and derivative domains

Question 1 of 8

No calculator

Find exact derivative terms, classify the origin and evaluate the specified gradient

f(x)=(2)x^{\frac{2}{3}}+(-3)

Use real roots and the maximal real domain of the printed expression. Here the ordinary derivative requires a finite two-sided derivative on an open real neighbourhood. A right derivative at an endpoint is stated separately

Use the nonconstant terms in descending exponent order after simplifying; constants differentiate to zero

x=27

(a)3 marks

For each nonconstant term, enter its original exponent n, then the derivative coefficient a and exponent e. List terms in descending original exponent order

\frac{d}{dx}(c_i x^{n_i})=a_i x^{e_i}

\(n_1\) \(a_1\) \(e_1\)

(b)1 mark

Select the function domain and ordinary derivative domain for the original printed expression

D_f=[0,\infty),\quad D_{f\prime}=[0,\infty)

D_f=\mathbb R\setminus\{0\},\quad D_{f\prime}=\mathbb R\setminus\{0\}

D_f=\mathbb R,\quad D_{f\prime}=\mathbb R\setminus\{0\}

D_f=\mathbb R,\quad D_{f\prime}=\mathbb R

Which statement about zero follows for this original function under the stated convention?

The function is defined at zero, but the admissible difference quotient has no finite derivative value there

The ordinary two-sided derivative at zero exists and equals zero

The original function is undefined at zero, so it has no derivative there

The right derivative at zero is zero, but the ordinary two-sided derivative is not defined there

(d)1 mark

Evaluate the derivative at the printed input. Enter an exact fraction if needed

f'(27)

Exact derivative value