If `y = sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))" then "(dy)/(dx)` is equal to

Options

A. ` (1)/(2) sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
`{(1)/(x-3) +(1)/(x-2) -(1)/(x-3)-(1)/(x-4) -(1)/(x-5)}`
B. ` (1)/(2) sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
`{(1)/(x-3) -(1)/(x-2) +(1)/(x-3)+(1)/(x-4) +(1)/(x-5)}`
C. ` (1)/(2) sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))`
`{(1)/(x-3) -(1)/(x-2) +(1)/(x-3)-(1)/(x-4) +(1)/(x-5)}`
D. None of the above

Explanation

To find the derivative of the function \( y = \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}} \), we will follow these steps:

### Step 1: Rewrite the Function
We can rewrite the function in a more manageable form:
\[ y = \left( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \right)^{1/2} \]

### Step 2: Apply the Chain Rule
To differentiate \( y \), we will use the chain rule. The derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{2} \left( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \right)^{-1/2} \cdot \frac{d}{dx} \left( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \right) \]

### Step 3: Differentiate the Quotient
Now we need to differentiate the quotient \( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \) using the quotient rule:
\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \]
where \( u = (x-1)(x-2) \) and \( v = (x-3)(x-4)(x-5) \).

#### Step 3.1: Differentiate \( u \)
\[ u = (x-1)(x-2) \implies u' = (x-1) + (x-2) = 2x - 3 \]

#### Step 3.2: Differentiate \( v \)
\[ v = (x-3)(x-4)(x-5) \]
Using the product rule:
\[ v' = (x-4)(x-5) + (x-3)(x-5) + (x-3)(x-4) = 3x^2 - 36x + 60 \]

### Step 4: Substitute \( u \), \( u' \), \( v \), and \( v' \)
Now substituting \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule:
\[ \frac{d}{dx} \left( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \right) = \frac{(2x - 3)(x-3)(x-4)(x-5) - (x-1)(x-2)(3x^2 - 36x + 60)}{((x-3)(x-4)(x-5))^2} \]

### Step 5: Combine Everything
Now substituting back into our derivative expression:
\[ \frac{dy}{dx} = \frac{1}{2} \left( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \right)^{-1/2} \cdot \frac{(2x - 3)(x-3)(x-4)(x-5) - (x-1)(x-2)(3x^2 - 36x + 60)}{((x-3)(x-4)(x-5))^2} \]

### Final Step: Simplify
This expression can be simplified further, but the key derivative is captured in the expression above.