To find the derivative of 
𝑓
(
𝑥
)
=
𝑥
2
+
2
𝑥
f(x)=x 
2
 +2x using the first principle of derivatives

To find the derivative of \( f(x) = x^2 + 2x \) using the first principle of derivatives, we use the definition of the derivative:

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

### Step-by-step process:

1. **Substitute \( f(x) = x^2 + 2x \) into the definition:**

   We first calculate \( f(x + h) \).

   \[
   f(x + h) = (x + h)^2 + 2(x + h)
   \]

   Expanding this:

   \[
   f(x + h) = (x^2 + 2xh + h^2) + 2x + 2h
   \]

   Simplifying:

   \[
   f(x + h) = x^2 + 2xh + h^2 + 2x + 2h
   \]

2. **Set up the difference \( f(x + h) - f(x) \):**

   \[
   f(x + h) - f(x) = (x^2 + 2xh + h^2 + 2x + 2h) - (x^2 + 2x)
   \]

   Simplifying this expression:

   \[
   f(x + h) - f(x) = x^2 + 2xh + h^2 + 2x + 2h - x^2 - 2x
   \]

   The \( x^2 \) and \( 2x \) terms cancel out, so we're left with:

   \[
   f(x + h) - f(x) = 2xh + h^2 + 2h
   \]

3. **Divide by \( h \):**

   Now, divide the difference by \( h \):

   \[
   \frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2 + 2h}{h}
   \]

   Simplifying this:

   \[
   \frac{f(x + h) - f(x)}{h} = 2x + h + 2
   \]

4. **Take the limit as \( h \to 0 \):**

   \[
   f'(x) = \lim_{h \to 0} (2x + h + 2) = 2x + 2
   \]

Thus, the derivative of \( f(x) = x^2 + 2x \) is:

\[
f'(x) = 2x + 2
\]

Would you like more details or have any questions?

### Relative Questions:
1. How does the first principle of derivatives work in general?
2. Can you apply the first principle to find the derivative of \( f(x) = x^3 \)?
3. What is the geometric interpretation of the derivative?
4. How does the first principle differ from the power rule for differentiation?
5. How would you find the second derivative of \( f(x) = x^2 + 2x \)?

### Tip:
Using the first principle is great for understanding the fundamental concept of derivatives, but for quicker calculations, mastering the differentiation rules (like the power rule) is crucial.

To find the derivative of f(x) = x^2 + 2x using the first principle of derivatives

Explore how to calculate the derivative of f(x) = x^2 + 2x using the first principle of derivatives. This detailed step-by-step guide covers fundamental calculus concepts and the application of the limit definition for differentiation. Suitable for students in Grade 11-12 or early university courses.

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To find the derivative of 𝑓 ( 𝑥 )

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