differentiate the function. y = (4x^2 + 6x + 2)/sqrt(x) y =

Explanation:

Step1: Rewrite the function

Rewrite $y=\frac{4x^{2}+6x + 2}{\sqrt{x}}$ as $y = 4x^{\frac{3}{2}}+6x^{\frac{1}{2}}+2x^{-\frac{1}{2}}$ using the rule $\frac{a^{m}}{a^{n}}=a^{m - n}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y'=nax^{n - 1}$. For $y_1 = 4x^{\frac{3}{2}}$, $y_1'=4\times\frac{3}{2}x^{\frac{3}{2}-1}=6x^{\frac{1}{2}}$. For $y_2 = 6x^{\frac{1}{2}}$, $y_2'=6\times\frac{1}{2}x^{\frac{1}{2}-1}=3x^{-\frac{1}{2}}$. For $y_3 = 2x^{-\frac{1}{2}}$, $y_3'=2\times(-\frac{1}{2})x^{-\frac{1}{2}-1}=-x^{-\frac{3}{2}}$.

Step3: Find the derivative of the function

$y'=y_1'+y_2'+y_3'=6x^{\frac{1}{2}}+3x^{-\frac{1}{2}}-x^{-\frac{3}{2}}$.

Answer:

$6\sqrt{x}+\frac{3}{\sqrt{x}}-\frac{1}{x\sqrt{x}}$