`y=(x+1)^(2)(x+2)^(3)(x+3)^(4)`

Answer

To find the derivative of the function \( y = (x+1)^2 (x+2)^3 (x+3)^4 \), we will use logarithmic differentiation. Here’s the step-by-step solution:

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of \( y \):
\[ \ln y = \ln \left( (x+1)^2 (x+2)^3 (x+3)^4 \right) \]

### Step 2: Apply the properties of logarithms
Using the properties of logarithms, we can expand the right-hand side:
\[ \ln y = 2 \ln(x+1) + 3 \ln(x+2) + 4 \ln(x+3) \]

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

### Step 4: Multiply both sides by \( y \)
To isolate \( \frac{dy}{dx} \), we multiply both sides by \( y \):
\[ \frac{dy}{dx} = y \left( 2 \cdot \frac{1}{x+1} + 3 \cdot \frac{1}{x+2} + 4 \cdot \frac{1}{x+3} \right) \]

### Step 5: Substitute back for \( y \)
Now we substitute back the expression for \( y \):
\[ \frac{dy}{dx} = (x+1)^2 (x+2)^3 (x+3)^4 \left( 2 \cdot \frac{1}{x+1} + 3 \cdot \frac{1}{x+2} + 4 \cdot \frac{1}{x+3} \right) \]

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

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