find `dy/dx` if `y = (x^(2) - 4x+5) (x^(3) - 2)`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = (x^2 - 4x + 5)(x^3 - 2)\), we will use the product rule of differentiation. The product rule states that if you have two functions \(u(x)\) and \(v(x)\), 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 the functions**:
Let:
\[ u = x^2 - 4x + 5 \]
\[ v = x^3 - 2 \]

2. **Differentiate \(u\) and \(v\)**:
- Differentiate \(u\):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2 - 4x + 5) = 2x - 4 \]
- Differentiate \(v\):
\[ \frac{dv}{dx} = \frac{d}{dx}(x^3 - 2) = 3x^2 \]

3. **Apply the product rule**:
Using the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substitute \(u\), \(v\), \(\frac{du}{dx}\), and \(\frac{dv}{dx}\):
\[ \frac{dy}{dx} = (x^2 - 4x + 5)(3x^2) + (x^3 - 2)(2x - 4) \]

4. **Expand the expression**:
- First term:
\[ (x^2 - 4x + 5)(3x^2) = 3x^4 - 12x^3 + 15x^2 \]
- Second term:
\[ (x^3 - 2)(2x - 4) = 2x^4 - 4x^3 - 4 + 8 = 2x^4 - 4x^3 + 8 \]

5. **Combine like terms**:
Now combine the results from the first and second terms:
\[ \frac{dy}{dx} = (3x^4 + 2x^4) + (-12x^3 - 4x^3) + (15x^2 + 8) \]
\[ = 5x^4 - 16x^3 + 15x^2 + 8 \]

Thus, the final result is:
\[ \frac{dy}{dx} = 5x^4 - 16x^3 + 15x^2 + 8 \]