Differentiate `sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))` with respect to `x`

Explanation

To differentiate the function \( y = \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}} \) with respect to \( x \), we will follow these steps:

### Step 1: Rewrite the Function
We start by rewriting 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 Logarithmic Differentiation
Taking the natural logarithm of both sides:
\[ \log y = \frac{1}{2} \log \left( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \right) \]

### Step 3: Use Logarithm Properties
Using the properties of logarithms:
\[ \log y = \frac{1}{2} \left( \log((x-1)(x-2)) - \log((x-3)(x-4)(x-5)) \right) \]
This can be expanded further:
\[ \log y = \frac{1}{2} \left( \log(x-1) + \log(x-2) - (\log(x-3) + \log(x-4) + \log(x-5)) \right) \]

### Step 4: Differentiate Both Sides
Now, we differentiate both sides with respect to \( x \):
\[ \frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \left( \frac{1}{x-1} + \frac{1}{x-2} - \left( \frac{1}{x-3} + \frac{1}{x-4} + \frac{1}{x-5} \right) \right) \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Multiplying both sides by \( y \):
\[ \frac{dy}{dx} = y \cdot \frac{1}{2} \left( \frac{1}{x-1} + \frac{1}{x-2} - \left( \frac{1}{x-3} + \frac{1}{x-4} + \frac{1}{x-5} \right) \right) \]

### Step 6: Substitute Back for \( y \)
Substituting \( y \) back into the equation:
\[ \frac{dy}{dx} = \frac{1}{2} \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}} \left( \frac{1}{x-1} + \frac{1}{x-2} - \left( \frac{1}{x-3} + \frac{1}{x-4} + \frac{1}{x-5} \right) \right) \]

### Final Answer
Thus, the derivative of the given function is:
\[ \frac{dy}{dx} = \frac{1}{2} \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}} \left( \frac{1}{x-1} + \frac{1}{x-2} - \frac{1}{x-3} - \frac{1}{x-4} - \frac{1}{x-5} \right) \]