Find the first and second derivatives of the function. $$ y=\frac{(x+4)\left(x^{2}-4 x+16\right)}{x^{3}} $$

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To find the first and second derivatives of the function:

$$
y = \frac{(x + 4)(x^2 - 4x + 16)}{x^3}
$$

**Step 1: Simplify the Function**

First, simplify the expression for $$ y $$.

$$
y = \frac{(x + 4)(x^2 - 4x + 16)}{x^3}
$$

Expand the numerator:

$$
(x + 4)(x^2 - 4x + 16) = x^3 - 4x^2 + 16x + 4x^2 - 16x + 64 = x^3 + 64
$$

So, the function simplifies to:

$$
y = \frac{x^3 + 64}{x^3} = 1 + \frac{64}{x^3} = 1 + 64x^{-3}
$$

**Step 2: Find the First Derivative ($$ y' $$)**

Differentiate $$ y $$ with respect to $$ x $$:

$$
y = 1 + 64x^{-3}
$$

$$
y' = \frac{d}{dx}[1] + \frac{d}{dx}[64x^{-3}] = 0 + 64 \cdot (-3)x^{-4} = -192x^{-4} = -\frac{192}{x^4}
$$

**Step 3: Find the Second Derivative ($$ y'' $$)**

Differentiate $$ y' $$ with respect to $$ x $$:

$$
y' = -192x^{-4}
$$

$$
y'' = \frac{d}{dx}[-192x^{-4}] = -192 \cdot (-4)x^{-5} = 768x^{-5} = \frac{768}{x^5}
$$

**Final Answer:**

$$
y' = -\frac{192}{x^4} \quad \text{and} \quad y'' = \frac{768}{x^5}
$$
The first derivative $$ y' $$ is $$ -\frac{192}{x^4} $$ and the second derivative $$ y'' $$ is $$ \frac{768}{x^5} $$.

Find the first and second derivatives of the function. \[ y=\frac{(x+4)\left(x^{2}-4 x+16\right)}{x^{3}} \]

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