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get the derivatives the following problem using increment method of differentiation

To solve these problems using the **increment method of differentiation**, also known as the **difference quotient method**, we use the definition of the derivative:

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

I'll walk you through the steps to solve each problem, starting with the first. Let's tackle these one by one.

### 1. \( f(x) = 3 - 2x + 4x^2 \)
Using the difference quotient method:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]
- First, find \( f(x+h) \):
\[
f(x+h) = 3 - 2(x+h) + 4(x+h)^2 = 3 - 2x - 2h + 4(x^2 + 2xh + h^2)
\]
\[
= 3 - 2x - 2h + 4x^2 + 8xh + 4h^2
\]

Now, compute the difference quotient:
\[
f'(x) = \lim_{h \to 0} \frac{(3 - 2x - 2h + 4x^2 + 8xh + 4h^2) - (3 - 2x + 4x^2)}{h}
\]
\[
= \lim_{h \to 0} \frac{-2h + 8xh + 4h^2}{h}
\]
Simplify:
\[
f'(x) = \lim_{h \to 0} (-2 + 8x + 4h) = -2 + 8x
\]
Thus, \( f'(x) = -2 + 8x \).

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If you want, I can solve each derivative similarly, step by step. Would you like me to continue with the others, or would you prefer to solve them together?

Also, here are five questions to consider for expanding your understanding:
1. How does the increment method differ from other differentiation methods?
2. What is the geometric interpretation of the derivative?
3. Can you apply the difference quotient to trigonometric functions?
4. What happens if the limit in the difference quotient doesn't exist?
5. Why does the limit \( h \to 0 \) lead to the instantaneous rate of change?

**Tip:** Practice applying the difference quotient to different functions to strengthen your understanding of derivatives.

Determine the derivatives of the following using the increment method of differentiation.

Learn how to compute the derivatives of functions using the increment method of differentiation. This guide covers algebraic and trigonometric functions, applying the difference quotient and limit definitions to obtain exact derivatives. Suitable for high school students (Grades 11-12) studying calculus.

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