Differentiate the following functions w.r.t. x.
`f(x) = sqrt( 4-x), x lt 4`

Answer

To differentiate the function \( f(x) = \sqrt{4 - x} \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
We have the function:
\[ f(x) = \sqrt{4 - x} \]

### Step 2: Rewrite the function using exponent notation
We can rewrite the square root in exponent form:
\[ f(x) = (4 - x)^{1/2} \]

### Step 3: Apply the chain rule
To differentiate \( f(x) \), we will use the chain rule. The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is:
\[ \frac{d}{dx} g(h(x)) = g'(h(x)) \cdot h'(x) \]

Here, let \( g(u) = u^{1/2} \) where \( u = 4 - x \).

### Step 4: Differentiate the outer function
The derivative of \( g(u) = u^{1/2} \) is:
\[ g'(u) = \frac{1}{2} u^{-1/2} = \frac{1}{2\sqrt{u}} \]

### Step 5: Differentiate the inner function
Now, differentiate the inner function \( h(x) = 4 - x \):
\[ h'(x) = -1 \]

### Step 6: Combine using the chain rule
Now, we can combine these results using the chain rule:
\[ f'(x) = g'(h(x)) \cdot h'(x) = \frac{1}{2\sqrt{4 - x}} \cdot (-1) \]

### Step 7: Simplify the expression
Thus, we have:
\[ f'(x) = -\frac{1}{2\sqrt{4 - x}} \]

### Final Answer
The derivative of the function \( f(x) = \sqrt{4 - x} \) with respect to \( x \) is:
\[ f'(x) = -\frac{1}{2\sqrt{4 - x}} \]

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