Differentiate the following w.r.t. x :
`sqrt((x-1)(x-2)(x-3)(x-4))`

Answer

To differentiate the function \( y = \sqrt{(x-1)(x-2)(x-3)(x-4)} \) with respect to \( x \), we will follow these steps:

### Step 1: Rewrite the Function
We can rewrite the function using exponent notation:
\[ y = \left( (x-1)(x-2)(x-3)(x-4) \right)^{1/2} \]

### Step 2: Apply the Chain Rule
To differentiate \( y \), we will use the chain rule. The derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{2} \left( (x-1)(x-2)(x-3)(x-4) \right)^{-1/2} \cdot \frac{d}{dx} \left( (x-1)(x-2)(x-3)(x-4) \right) \]

### Step 3: Differentiate the Inner Function
Next, we need to differentiate the product \( (x-1)(x-2)(x-3)(x-4) \). We will use the product rule. Let:
\[ u = (x-1), \quad v = (x-2), \quad w = (x-3), \quad z = (x-4) \]
Then,
\[ \frac{d}{dx} (uvwz) = u'v w z + uv'w z + uvw'z + uvwz' \]
Calculating each derivative:
- \( u' = 1 \)
- \( v' = 1 \)
- \( w' = 1 \)
- \( z' = 1 \)

Thus, we have:
\[ \frac{d}{dx} (uvwz) = (1)(x-2)(x-3)(x-4) + (x-1)(1)(x-3)(x-4) + (x-1)(x-2)(1)(x-4) + (x-1)(x-2)(x-3)(1) \]

### Step 4: Simplify the Derivative
Now, we can simplify the expression:
\[ = (x-2)(x-3)(x-4) + (x-1)(x-3)(x-4) + (x-1)(x-2)(x-4) + (x-1)(x-2)(x-3) \]

### Step 5: Substitute Back into the Derivative
Now we substitute this back into our derivative:
\[ \frac{dy}{dx} = \frac{1}{2} \left( (x-1)(x-2)(x-3)(x-4) \right)^{-1/2} \cdot \left[ (x-2)(x-3)(x-4) + (x-1)(x-3)(x-4) + (x-1)(x-2)(x-4) + (x-1)(x-2)(x-3) \right] \]

### Step 6: Final Expression
Thus, we can express the final derivative as:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{(x-1)(x-2)(x-3)(x-4)}} \cdot \left[ (x-2)(x-3)(x-4) + (x-1)(x-3)(x-4) + (x-1)(x-2)(x-4) + (x-1)(x-2)(x-3) \right] \]