What is the rate of change of `sqrt(x^(2)+16)` w.r.t. `x^(2)` at `x=3` ?

Options

A. `1/5`
B. `(1)/(10)`
C. `(1)/(20)`
D. `(1)/(15)`

Answer

To find the rate of change of \( \sqrt{x^2 + 16} \) with respect to \( x^2 \) at \( x = 3 \), we will follow these steps:

### Step 1: Define the function
Let \( y = \sqrt{x^2 + 16} \).

### Step 2: Differentiate with respect to \( x^2 \)
To find the rate of change of \( y \) with respect to \( x^2 \), we will use the chain rule. We need to find \( \frac{dy}{dx} \) and \( \frac{dx^2}{dx} \).

1. Differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x^2 + 16}} \cdot \frac{d}{dx}(x^2 + 16) = \frac{1}{2\sqrt{x^2 + 16}} \cdot 2x = \frac{x}{\sqrt{x^2 + 16}}. \]

2. Differentiate \( x^2 \) with respect to \( x \):
\[ \frac{dx^2}{dx} = 2x. \]

### Step 3: Use the chain rule
Now, we can find \( \frac{dy}{dx^2} \) using the chain rule:
\[ \frac{dy}{dx^2} = \frac{dy}{dx} \cdot \frac{dx}{dx^2} = \frac{dy}{dx} \cdot \frac{1}{\frac{dx^2}{dx}} = \frac{dy}{dx} \cdot \frac{1}{2x}. \]
Substituting \( \frac{dy}{dx} \):
\[ \frac{dy}{dx^2} = \frac{x}{\sqrt{x^2 + 16}} \cdot \frac{1}{2x} = \frac{1}{2\sqrt{x^2 + 16}}. \]

### Step 4: Evaluate at \( x = 3 \)
Now we substitute \( x = 3 \):
\[ \frac{dy}{dx^2} \bigg|_{x=3} = \frac{1}{2\sqrt{3^2 + 16}} = \frac{1}{2\sqrt{9 + 16}} = \frac{1}{2\sqrt{25}} = \frac{1}{2 \cdot 5} = \frac{1}{10}. \]

### Final Answer
The rate of change of \( \sqrt{x^2 + 16} \) with respect to \( x^2 \) at \( x = 3 \) is \( \frac{1}{10} \).

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