Question: Create a math video explanation for a university-level calculus lesson about derivatives and their geometrical meaning.
Explain the function $f(x) = x^2 + 2x + 1$.
Find the derivative step by step, then calculate the slope of the tangent line at $x = 1$.
Explain each mathematical step in simple English, show the equations clearly, and include a visual explanation of the tangent line on the graph.
Include narration, a final answer, and a quick verification of the result.
1. **Problem Statement:** We are given the function $$f(x) = x^2 + 2x + 1$$ and need to find its derivative, then calculate the slope of the tangent line at $$x = 1$$.
2. **Recall the derivative definition:** The derivative $$f'(x)$$ represents the slope of the tangent line to the curve at any point $$x$$.
3. **Derivative rules used:**
- Power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$
- Sum rule: $$\frac{d}{dx} (u + v) = \frac{du}{dx} + \frac{dv}{dx}$$
4. **Calculate the derivative step-by-step:**
$$f(x) = x^2 + 2x + 1$$
Apply derivative term-wise:
$$f'(x) = \frac{d}{dx} x^2 + \frac{d}{dx} 2x + \frac{d}{dx} 1$$
Using power rule and constants:
$$f'(x) = 2x + 2 + 0$$
Simplify:
$$f'(x) = 2x + 2$$
5. **Calculate slope at $$x=1$$:**
Substitute $$x=1$$ into $$f'(x)$$:
$$f'(1) = 2(1) + 2 = 2 + 2 = 4$$
6. **Interpretation:** The slope of the tangent line to the curve at $$x=1$$ is $$4$$.
7. **Equation of tangent line:**
Use point-slope form:
$$y - f(1) = f'(1)(x - 1)$$
Calculate $$f(1)$$:
$$f(1) = 1^2 + 2(1) + 1 = 1 + 2 + 1 = 4$$
So,
$$y - 4 = 4(x - 1)$$
Simplify:
$$y = 4x - 4 + 4 = 4x$$
8. **Verification:**
Check that the tangent line touches the curve at $$x=1$$:
$$f(1) = 4$$ and tangent line at $$x=1$$ is $$y=4(1)=4$$, confirming correctness.
**Final answer:**
- Derivative: $$f'(x) = 2x + 2$$
- Slope at $$x=1$$: $$4$$
- Tangent line equation: $$y = 4x$$