`f(x)=(3x^(2)+2)^(3)(5x-1)^(2)` find `dy/dx`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( f(x) = (3x^2 + 2)^3 (5x - 1)^2 \), we will apply the product rule and the chain rule.

### Step-by-Step Solution:

1. **Identify the components**:
Let \( u = (3x^2 + 2)^3 \) and \( v = (5x - 1)^2 \). We need to differentiate \( y = u \cdot v \).

2. **Apply the product rule**:
The product rule states that \( \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \).

3. **Differentiate \( u \)**:
To differentiate \( u = (3x^2 + 2)^3 \), we use the chain rule:
\[ \frac{du}{dx} = 3(3x^2 + 2)^2 \cdot \frac{d}{dx}(3x^2 + 2) = 3(3x^2 + 2)^2 \cdot (6x) = 18x(3x^2 + 2)^2 \]

4. **Differentiate \( v \)**:
To differentiate \( v = (5x - 1)^2 \), we again use the chain rule:
\[ \frac{dv}{dx} = 2(5x - 1) \cdot \frac{d}{dx}(5x - 1) = 2(5x - 1) \cdot 5 = 10(5x - 1) \]

5. **Substitute back into the product rule**:
Now, substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the product rule:
\[ \frac{dy}{dx} = (3x^2 + 2)^3 \cdot 10(5x - 1) + (5x - 1)^2 \cdot 18x(3x^2 + 2)^2 \]

6. **Factor out common terms**:
Notice that \( (3x^2 + 2)^2 \) and \( (5x - 1) \) are common in both terms:
\[ \frac{dy}{dx} = (3x^2 + 2)^2 (5x - 1) \left[ 10(3x^2 + 2) + 18x(5x - 1) \right] \]

7. **Simplify the expression inside the brackets**:
Now, we simplify \( 10(3x^2 + 2) + 18x(5x - 1) \):
\[ = 30x^2 + 20 + 90x^2 - 18x = 120x^2 - 18x + 20 \]

8. **Final derivative**:
Thus, the final expression for the derivative is:
\[ \frac{dy}{dx} = (3x^2 + 2)^2 (5x - 1)(120x^2 - 18x + 20) \]