Find `dy/dx` for each of the following
`y=(x^(2)+3x+5)(x^(2)-2)^(2)`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = (x^2 + 3x + 5)(x^2 - 2)^2\), we will use the product rule of differentiation. The product rule states that if you have two functions \(u\) and \(v\), then the derivative of their product is given by:

\[ \frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step-by-step Solution:

1. **Identify \(u\) and \(v\)**:
Let:
\[ u = x^2 + 3x + 5 \]
\[ v = (x^2 - 2)^2 \]

2. **Differentiate \(u\)**:
To find \(\frac{du}{dx}\):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + 3x + 5) = 2x + 3 \]

3. **Differentiate \(v\)**:
To differentiate \(v\), we will use the chain rule:
\[ v = (x^2 - 2)^2 \]
Let \(w = x^2 - 2\), then \(v = w^2\). Using the chain rule:
\[ \frac{dv}{dx} = 2w \cdot \frac{dw}{dx} = 2(x^2 - 2)(2x) = 4x(x^2 - 2) \]

4. **Apply the product rule**:
Now, we can apply the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting the values we found:
\[ \frac{dy}{dx} = (x^2 + 3x + 5)(4x(x^2 - 2)) + ((x^2 - 2)^2)(2x + 3) \]

5. **Simplify the expression**:
First, simplify the first term:
\[ = 4x(x^2 - 2)(x^2 + 3x + 5) \]
And for the second term:
\[ = (x^2 - 2)^2(2x + 3) \]
Thus, we have:
\[ \frac{dy}{dx} = 4x(x^2 - 2)(x^2 + 3x + 5) + (x^2 - 2)^2(2x + 3) \]

6. **Factor out common terms**:
Notice that \((x^2 - 2)\) is a common factor:
\[ = (x^2 - 2)\left[4x(x^2 + 3x + 5) + (x^2 - 2)(2x + 3)\right] \]

### Final Result:
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = (x^2 - 2)\left[4x(x^2 + 3x + 5) + (x^2 - 2)(2x + 3)\right] \]