The rate of change of `sqrt(x^2+16)` with respect to `x/(x-1)` at `x=3` is

Options

A. 1
B. `(11)/(5)`
C. `-(12)/(5)`
D. `-3`

Explanation

To find the rate of change of \( \sqrt{x^2 + 16} \) with respect to \( \frac{x}{x-1} \) at \( x = 3 \), we will follow these steps:

### Step 1: Define the Functions
Let:
- \( y = \sqrt{x^2 + 16} \)
- \( z = \frac{x}{x-1} \)

### Step 2: Differentiate \( y \) with respect to \( x \)
Using the chain rule, we differentiate \( y \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x^2 + 16}} \cdot (2x) = \frac{x}{\sqrt{x^2 + 16}} \]

### Step 3: Differentiate \( z \) with respect to \( x \)
To differentiate \( z \), we can use the quotient rule:
\[ z = \frac{x}{x-1} \]
Using the quotient rule, we have:
\[ \frac{dz}{dx} = \frac{(x-1)(1) - x(1)}{(x-1)^2} = \frac{x - 1 - x}{(x-1)^2} = \frac{-1}{(x-1)^2} \]

### Step 4: Find \( \frac{dy}{dz} \)
Using the relationship:
\[ \frac{dy}{dz} = \frac{dy/dx}{dz/dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dz} = \frac{\frac{x}{\sqrt{x^2 + 16}}}{\frac{-1}{(x-1)^2}} = -\frac{x (x-1)^2}{\sqrt{x^2 + 16}} \]

### Step 5: Evaluate at \( x = 3 \)
Now we substitute \( x = 3 \):
\[ \frac{dy}{dz} \bigg|_{x=3} = -\frac{3(3-1)^2}{\sqrt{3^2 + 16}} = -\frac{3 \cdot 4}{\sqrt{9 + 16}} = -\frac{12}{\sqrt{25}} = -\frac{12}{5} \]

### Final Answer
The rate of change of \( \sqrt{x^2 + 16} \) with respect to \( \frac{x}{x-1} \) at \( x = 3 \) is:
\[ \boxed{-\frac{12}{5}} \]