If `f(x) = (x-4) / (2sqrtx)`, then `f' (4)` is equal to

Answer

To find \( f'(4) \) for the function \( f(x) = \frac{x-4}{2\sqrt{x}} \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form of \( \frac{u}{v} \), then its derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

### Step 1: Identify \( u \) and \( v \)
In our case:
- \( u = x - 4 \)
- \( v = 2\sqrt{x} \)

### Step 2: Differentiate \( u \) and \( v \)
Now we differentiate \( u \) and \( v \):
- \( \frac{du}{dx} = 1 \)
- To differentiate \( v = 2\sqrt{x} \), we use the chain rule:
\[ \frac{dv}{dx} = 2 \cdot \frac{1}{2\sqrt{x}} \cdot \frac{d}{dx}(x) = \frac{1}{\sqrt{x}} \]

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:
\[ f'(x) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ f'(x) = \frac{(2\sqrt{x})(1) - (x - 4)\left(\frac{1}{\sqrt{x}}\right)}{(2\sqrt{x})^2} \]

### Step 4: Simplify the Expression
Now we simplify the expression:
\[ f'(x) = \frac{2\sqrt{x} - \frac{x - 4}{\sqrt{x}}}{4x} \]
To combine the terms in the numerator, we can write:
\[ f'(x) = \frac{2\sqrt{x} \cdot \sqrt{x} - (x - 4)}{4x} = \frac{2x - x + 4}{4x} = \frac{x + 4}{4x} \]

### Step 5: Evaluate at \( x = 4 \)
Now we need to find \( f'(4) \):
\[ f'(4) = \frac{4 + 4}{4 \cdot 4} = \frac{8}{16} = \frac{1}{2} \]

### Final Answer
Thus, \( f'(4) = \frac{1}{2} \).

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