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

To find the first and second derivatives of the function \[ y=\frac{(x+4)\left(x^{2}-4 x+16\right)}{x^{3}}, \] we'll first simplify the function before taking derivatives. ### Step 1: Simplification Let’s simplify the function first. \[ y = \frac{(x+4)(x^2 - 4x + 16)}{x^3} \] Expanding the numerator: \[ (x + 4)(x^2 - 4x + 16) = x^3 - 4x^2 + 16x + 4x^2 - 16x + 64 = x^3 + 64. \] Now, rewriting the function gives us: \[ y = \frac{x^3 + 64}{x^3} = 1 + \frac{64}{x^3} = 1 + 64x^{-3}. \] ### Step 2: First Derivative Now we differentiate \( y \). \[ \frac{dy}{dx} = 0 - 64 \cdot (-3)x^{-4} = \frac{192}{x^4}. \] ### Step 3: Second Derivative Next, differentiate the first derivative to find the second derivative. \[ \frac{d^2y}{dx^2} = 192 \cdot (-4)x^{-5} = -\frac{768}{x^5}. \] ### Conclusion The first derivative is \[ \frac{dy}{dx} = \frac{192}{x^4}, \] and the second derivative is \[ \frac{d^2y}{dx^2} = -\frac{768}{x^5}. \]