Question

therefore, we have
$y = \frac{9x^2 + 3x + 3}{\sqrt{x}} = 9x^{3/2} + 3x^{1/2} + 3x^{-1/2}$.
now, since the function is a sum of power functions, we can use the power rule to find its derivative. recall that according to the power rule, the derivative of $x^n$, where $n$ is any real number, is $nx^{n - 1}$. so use the power rule, we have to calculate the following.
$\frac{3}{2} - 1 = \square$
$\frac{1}{2} - 1 = \square$
$-\frac{1}{2} - 1 = \square$
applying the power rule we have
$y = \square x^{1/2} + \square x^{-1/2} + \left(\square\
ight)x^{-3/2}$.
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Explanation:

Step1: Simplify the first term

$\frac{3}{2} - 1 = \frac{1}{2}$

Step2: Simplify the second term

$\frac{1}{2} - 1 = -\frac{1}{2}$

Step3: Simplify the third term

$-\frac{1}{2} - 1 = -\frac{3}{2}$

Step4: Apply the Power Rule to each term

$y' = 9 \times \frac{3}{2}x^{1/2} + 3 \times \frac{1}{2}x^{-1/2} + 3 \times (-\frac{1}{2})x^{-3/2}$

Step5: Calculate coefficients

$9 \times \frac{3}{2} = \frac{27}{2}$, $3 \times \frac{1}{2} = \frac{3}{2}$, $3 \times -\frac{1}{2} = -\frac{3}{2}$

Answer:

First blank: $\frac{1}{2}$
Second blank: $-\frac{1}{2}$
Third blank: $-\frac{3}{2}$
Fourth blank: $\frac{27}{2}$
Fifth blank: $\frac{3}{2}$
Sixth blank: $-\frac{3}{2}$
Final derivative: $\frac{27}{2}x^{1/2} + \frac{3}{2}x^{-1/2} - \frac{3}{2}x^{-3/2}$