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To solve this question, we will need to find the derivatives of the given functions using the definition of limits. The definition of the derivative, using limits, is:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

We will apply this to each of the given functions.

### (i) \( f(x) = 3x - 2 \)

\[
f'(x) = \lim_{h \to 0} \frac{(3(x+h) - 2) - (3x - 2)}{h}
\]
Simplifying the expression:
\[
f'(x) = \lim_{h \to 0} \frac{(3x + 3h - 2) - (3x - 2)}{h}
\]
\[
f'(x) = \lim_{h \to 0} \frac{3h}{h} = \lim_{h \to 0} 3 = 3
\]

Thus, the derivative of \( f(x) = 3x - 2 \) is \( f'(x) = 3 \).

### (ii) \( f(x) = 6 - 3x + 2x^2 \)

\[
f'(x) = \lim_{h \to 0} \frac{(6 - 3(x+h) + 2(x+h)^2) - (6 - 3x + 2x^2)}{h}
\]
Expand and simplify:
\[
f(x+h) = 6 - 3(x+h) + 2(x+h)^2 = 6 - 3x - 3h + 2(x^2 + 2xh + h^2) = 6 - 3x - 3h + 2x^2 + 4xh + 2h^2
\]
So,
\[
f'(x) = \lim_{h \to 0} \frac{(6 - 3x - 3h + 2x^2 + 4xh + 2h^2) - (6 - 3x + 2x^2)}{h}
\]
Simplifying further:
\[
f'(x) = \lim_{h \to 0} \frac{- 3h + 4xh + 2h^2}{h} = \lim_{h \to 0} (-3 + 4x + 2h)
\]
\[
f'(x) = -3 + 4x
\]

Thus, the derivative of \( f(x) = 6 - 3x + 2x^2 \) is \( f'(x) = -3 + 4x \).

### (iii) \( f(x) = x^3 + 5x^2 + 6x + 2 \)

\[
f'(x) = \lim_{h \to 0} \frac{((x+h)^3 + 5(x+h)^2 + 6(x+h) + 2) - (x^3 + 5x^2 + 6x + 2)}{h}
\]
Expand and simplify:
\[
f(x+h) = (x+h)^3 + 5(x+h)^2 + 6(x+h) + 2 = x^3 + 3x^2h + 3xh^2 + h^3 + 5(x^2 + 2xh + h^2) + 6x + 6h + 2
\]
So:
\[
f'(x) = \lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3 + 5x^2 + 10xh + 5h^2 + 6x + 6h + 2) - (x^3 + 5x^2 + 6x + 2)}{h}
\]
Simplify:
\[
f'(x) = \lim_{h \to 0} \frac{(3x^2h + 10xh + 6h + 3xh^2 + 5h^2 + h^3)}{h}
\]
\[
f'(x) = \lim_{h \to 0} (3x^2 + 10x + 6 + 3xh + 5h + h^2)
\]
As \( h \to 0 \), the terms with \( h \) vanish:
\[
f'(x) = 3x^2 + 10x + 6
\]

Thus, the derivative of \( f(x) = x^3 + 5x^2 + 6x + 2 \) is \( f'(x) = 3x^2 + 10x + 6 \).

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Would you like more details on any of these steps or any other clarification?

### Here are 5 related questions:
1. How does the derivative relate to the slope of the tangent line to the curve?
2. Can the power rule be derived from the definition of a limit?
3. What is the geometric interpretation of the derivative?
4. How do higher-order derivatives relate to the original function?
5. How can you apply the definition of the derivative to a trigonometric function?

### Tip:
When using the limit definition of the derivative, it's helpful to carefully expand and simplify expressions step by step to avoid mistakes.

Using the definition of limits, find the derivative of the following functions: (i) 3x - 2, (ii) 6 - 3x + 2x^2, (iii) x^3 + 5x^2 + 6x + 2.

Learn how to derive the derivatives of polynomial functions such as 3x - 2, 6 - 3x + 2x^2, and x^3 + 5x^2 + 6x + 2 using the limit definition of a derivative. This problem is suitable for high school calculus students and includes step-by-step solutions.

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