Find `(dy)/(dx)`, when:
`y=(x^(2)sqrt(1+x))/((1+x^(2))^(3//2))`

Answer

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

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

we will use logarithmic differentiation to simplify the process. Here are the steps:

### Step 1: Take the natural logarithm of both sides
We start by taking the logarithm of both sides:

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

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

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

This simplifies to:

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

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

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

### Step 4: Simplify the right-hand side
This gives us:

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

### Step 5: Multiply by \(y\) to isolate \(\frac{dy}{dx}\)
Now, we multiply both sides by \(y\):

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

### Step 6: Substitute back for \(y\)
Substituting back the expression for \(y\):

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

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

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

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