`y=((x+1)^(2)cdot sqrt(x-1))/((x+3)^(3)e^(x))`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function

\[ y = \frac{(x+1)^2 \sqrt{x-1}}{(x+3)^3 e^x}, \]

we will use logarithmic differentiation. Here’s a step-by-step solution:

### Step 1: Take the natural logarithm of both sides
Taking the natural logarithm helps simplify the differentiation process:

\[ \ln y = \ln\left(\frac{(x+1)^2 \sqrt{x-1}}{(x+3)^3 e^x} \right). \]

### Step 2: Apply the properties of logarithms
Using the properties of logarithms, we can separate the terms:

\[ \ln y = \ln((x+1)^2) + \ln(\sqrt{x-1}) - \ln((x+3)^3) - \ln(e^x). \]

This simplifies to:

\[ \ln y = 2 \ln(x+1) + \frac{1}{2} \ln(x-1) - 3 \ln(x+3) - x. \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides using implicit differentiation:

\[ \frac{1}{y} \frac{dy}{dx} = \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+3} - 1. \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Multiply both sides by \( y \) to isolate \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = y \left( \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+3} - 1 \right). \]

### Step 5: Substitute back for \( y \)
Now substitute back the expression for \( y \):

\[ \frac{dy}{dx} = \frac{(x+1)^2 \sqrt{x-1}}{(x+3)^3 e^x} \left( \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+3} - 1 \right). \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{(x+1)^2 \sqrt{x-1}}{(x+3)^3 e^x} \left( \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+3} - 1 \right). \]

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