If `f(x)=(x-4)/(2sqrt(x))`, then find `f'(1)`

Answer

To find \( f'(1) \) for the function \( f(x) = \frac{x - 4}{2\sqrt{x}} \), we will follow these steps:

### Step 1: Differentiate \( f(x) \)
We will use the quotient rule for differentiation, which states that if \( f(x) = \frac{u}{v} \), then:
\[ f'(x) = \frac{v \cdot u' - u \cdot v'}{v^2} \]
where \( u = x - 4 \) and \( v = 2\sqrt{x} \).

### Step 2: Identify \( u \) and \( v \)
- \( u = x - 4 \)
- \( v = 2\sqrt{x} \)

### Step 3: Find \( u' \) and \( v' \)
- \( u' = \frac{d}{dx}(x - 4) = 1 \)
- \( v' = \frac{d}{dx}(2\sqrt{x}) = 2 \cdot \frac{1}{2\sqrt{x}} = \frac{1}{\sqrt{x}} \)

### Step 4: Apply the Quotient Rule
Now, substituting \( u \), \( v \), \( u' \), and \( v' \) into the quotient rule formula:
\[ f'(x) = \frac{(2\sqrt{x}) \cdot (1) - (x - 4) \cdot \left(\frac{1}{\sqrt{x}}\right)}{(2\sqrt{x})^2} \]
This simplifies to:
\[ f'(x) = \frac{2\sqrt{x} - \frac{x - 4}{\sqrt{x}}}{4x} \]

### Step 5: Simplify the Numerator
To simplify the numerator:
\[ f'(x) = \frac{2\sqrt{x} - \frac{x - 4}{\sqrt{x}}}{4x} = \frac{2\sqrt{x} - \frac{x}{\sqrt{x}} + \frac{4}{\sqrt{x}}}{4x} \]
This becomes:
\[ f'(x) = \frac{2\sqrt{x} - \sqrt{x} + \frac{4}{\sqrt{x}}}{4x} = \frac{\sqrt{x} + \frac{4}{\sqrt{x}}}{4x} \]

### Step 6: Combine the Terms
Now, express \( \frac{4}{\sqrt{x}} \) as \( \frac{4}{\sqrt{x}} = \frac{4\sqrt{x}}{x} \):
\[ f'(x) = \frac{\sqrt{x} + \frac{4\sqrt{x}}{x}}{4x} = \frac{\sqrt{x}(1 + \frac{4}{x})}{4x} \]

### Step 7: Substitute \( x = 1 \)
Now, we substitute \( x = 1 \):
\[ f'(1) = \frac{\sqrt{1}(1 + \frac{4}{1})}{4 \cdot 1} = \frac{1(1 + 4)}{4} = \frac{5}{4} \]

### Final Answer
Thus, \( f'(1) = \frac{5}{4} \).

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