(i) Find the differentiation of `y= (x^(2) - 4x +5) (x^(3) -2)`

Answer

To find the differentiation of 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 \( u \) and \( v \), then the derivative of their product \( y = uv \) is given by:

\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

**Step 1: Identify the functions \( u \) and \( v \)**

Let:
- \( u = x^2 - 4x + 5 \)
- \( v = x^3 - 2 \)

**Step 2: Differentiate \( u \) and \( v \)**

Now, we will find the derivatives of \( u \) and \( v \):

1. Differentiate \( u \):
\[ \frac{du}{dx} = 2x - 4 \]

2. Differentiate \( v \):
\[ \frac{dv}{dx} = 3x^2 \]

**Step 3: Apply the product rule**

Using the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

Substituting the values of \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = (x^2 - 4x + 5)(3x^2) + (x^3 - 2)(2x - 4) \]

**Step 4: Expand the expression**

Now we will expand both terms:

1. Expanding the first term:
\[ (x^2 - 4x + 5)(3x^2) = 3x^4 - 12x^3 + 15x^2 \]

2. Expanding the second term:
\[ (x^3 - 2)(2x - 4) = 2x^4 - 4x^3 - 4x + 8 \]

**Step 5: Combine like terms**

Now, we will combine all the terms:
\[ \frac{dy}{dx} = (3x^4 - 12x^3 + 15x^2) + (2x^4 - 4x^3 - 4x + 8) \]

Combining like terms:
\[ = (3x^4 + 2x^4) + (-12x^3 - 4x^3) + 15x^2 - 4x + 8 \]
\[ = 5x^4 - 16x^3 + 15x^2 - 4x + 8 \]

**Final Result:**
Thus, the derivative of \( y \) is:
\[ \frac{dy}{dx} = 5x^4 - 16x^3 + 15x^2 - 4x + 8 \]

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