Find the derivative of f(x)=x²+2x using the first principle.

The derivative of a function \( f(x) \) using the first principle (or definition of the derivative) is given by:

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

Given \( f(x) = x^2 + 2x \), let's calculate the derivative step-by-step.

### Step 1: Compute \( f(x+h) \)
Substitute \( x+h \) into the function \( f(x) \):

\[
f(x+h) = (x+h)^2 + 2(x+h)
\]
Expanding the terms:
\[
f(x+h) = (x^2 + 2xh + h^2) + (2x + 2h)
\]
\[
f(x+h) = x^2 + 2xh + h^2 + 2x + 2h
\]

### Step 2: Set up the difference \( f(x+h) - f(x) \)
Now subtract \( f(x) = x^2 + 2x \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (x^2 + 2xh + h^2 + 2x + 2h) - (x^2 + 2x)
\]
Simplify the expression:
\[
f(x+h) - f(x) = x^2 + 2xh + h^2 + 2x + 2h - x^2 - 2x
\]
\[
f(x+h) - f(x) = 2xh + h^2 + 2h
\]

### Step 3: Divide by \( h \)
Next, divide the expression by \( h \):

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

### Step 4: Take the limit as \( h \to 0 \)
Finally, take the limit of the expression as \( h \to 0 \):

\[
f'(x) = \lim_{h \to 0} (2x + h + 2)
\]
Since \( h \to 0 \), we get:
\[
f'(x) = 2x + 2
\]

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

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

Would you like more details or have any questions?

### Relative questions:
1. How is the first principle of differentiation different from using standard rules?
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 would the first principle work for non-polynomial functions, like \( \sin(x) \)?
5. What are the limitations of using the first principle for complex functions?

### Tip:
The first principle of differentiation is a useful tool for understanding the concept of limits and how they apply to the slope of a function at a point.

Learn how to calculate the derivative of f(x) = x² + 2x using the first principle of differentiation. This step-by-step solution covers key calculus concepts, including limits and the first principle of differentiation. Suitable for students in Grades 11-12.

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