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 can follow these steps:

### Step 1: Rewrite the function
Let:
\[ y = \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}} \]
We can rewrite this as:
\[ y = \left( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \right)^{1/2} \]

### Step 2: Take the natural logarithm
Taking the natural logarithm of both sides:
\[ \ln y = \frac{1}{2} \ln \left( \frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)} \right) \]

### Step 3: Apply logarithmic properties
Using the properties of logarithms:
\[ \ln y = \frac{1}{2} \left( \ln(x-1) + \ln(x-2) - \ln(x-3) - \ln(x-4) - \ln(x-5) \right) \]

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

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

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

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

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