The exercise requires finding the derivative $\frac{dy}{dx}$ for each function listed. Here’s how we can approach each problem step-by-step:

Solutions:

I'll go over the first few derivatives in detail:

1. For $y = 2x^2 - 4x^2 + 7x - 5$:

   • Simplify the function first if possible. It appears there might be a mistake with two terms labeled $2x^2$ and $-4x^2$; they should combine.
   • If correct, find $\frac{dy}{dx} = 4x + 7$.

2. For $y = 9 - 3x^{-1} + 6x^{-2}$:

   • Differentiate each term separately.
   • $\frac{dy}{dx} = 0 + 3x^{-2} - 12x^{-3} = \frac{3}{x^2} - \frac{12}{x^3}$.

3. For $y = (x^2 - x)(2x - 6)$:

   • Expand and then differentiate each term.
   • First, expand $y = 2x^3 - 6x^2 - 2x^2 + 6x = 2x^3 - 8x^2 + 6x$.
   • Then, differentiate: $\frac{dy}{dx} = 6x^2 - 16x + 6$.

4. For $y = (3x^2)(2 + x^{-1})$:

   • Apply the product rule.
   • Let $u = 3x^2$ and $v = 2 + x^{-1}$.
   • Differentiate: $\frac{dy}{dx} = u'v + uv'$.

5. For $y = (x^{221} - 3x)(2 + x^{-1})$:

   • Use the product rule as in question 4.

Would you like solutions for each function, or just specific ones?

Practice Questions

1. How do you apply the product rule in differentiation?
2. What is the derivative of $x^n$ for any integer $n$?
3. How can we simplify expressions before differentiating?
4. Why is it important to combine like terms in polynomial expressions before differentiating?
5. How does the chain rule work in differentiating composite functions?

Tip

Always simplify the function as much as possible before differentiating to make the process easier and reduce errors.