Read the following information carefully and answer the question given below:
Consider the function `f(x)={{:(x^(2)-5,xle3),(sqrt(x+13),xgt3):}`
What is the differential coefficient of f(x) at x = 12?

Options

A. `(5)/(2)`
B. 5
C. `(1)/(5)`
D. `(1)/(10)`

Answer

To find the differential coefficient of the function \( f(x) \) at \( x = 12 \), we will follow these steps:

### Step 1: Identify the relevant piece of the function
The function \( f(x) \) is defined piecewise:
- For \( x \leq 3 \), \( f(x) = x^2 - 5 \)
- For \( x > 3 \), \( f(x) = \sqrt{x + 13} \)

Since we are interested in \( x = 12 \), which is greater than 3, we will use the second piece of the function:
\[ f(x) = \sqrt{x + 13} \]

### Step 2: Differentiate the function
To find the differential coefficient, we need to differentiate \( f(x) \):
\[ f(x) = (x + 13)^{1/2} \]
Using the power rule for differentiation:
\[ f'(x) = \frac{1}{2}(x + 13)^{-1/2} \cdot (1) = \frac{1}{2\sqrt{x + 13}} \]

### Step 3: Substitute \( x = 12 \) into the derivative
Now, we will substitute \( x = 12 \) into the derivative to find \( f'(12) \):
\[ f'(12) = \frac{1}{2\sqrt{12 + 13}} = \frac{1}{2\sqrt{25}} = \frac{1}{2 \cdot 5} = \frac{1}{10} \]

### Conclusion
Thus, the differential coefficient of \( f(x) \) at \( x = 12 \) is:
\[ \boxed{\frac{1}{10}} \]

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