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In calculus, the derivative is a fundamental concept that measures how a function changes as its input changes. For the given function, the derivative is found to determine the slope of the function at any point.
To find the derivative of a function like \( f(x) = x^2 + 2x \), we apply differentiation rules. Differentiation is the process of finding a derivative, using rules such as the power rule and sum rule.
The power rule states that for any function of the form \( ax^n \), the derivative is \( anx^{n-1} \). Applying this to our function, we differentiate each term:

• \( \frac{d}{dx}(x^2) = 2x \)
• \( \frac{d}{dx}(2x) = 2 \)

Adding these together gives us the derivative \( f'(x) = 2x + 2 \). This derivative helps in understanding the rate at which \( f(x) \) changes and is vital for finding the tangent line.

The tangent line to a curve at a given point is the straight line that touches the curve at that point and has the same slope as the curve.
When we consider the tangent line for a function, it gives us the best linear approximation to the curve at some specific point. With the function \( f(x) = x^2 + 2x \), the tangent line at \( a = 0 \) can be derived using the derivative.
A tangent line is given by the equation: \[ y = f(a) + f'(a)(x - a) \] where \( f(a) \) is the function value at \( a \) and \( f'(a) \) is the derivative at \( a \).
For our chosen point \( a = 0 \):

• \( f(0) = 0 \)
• \( f'(0) = 2 \)

Substituting these values, the tangent line equation is \[ y = 0 + 2(x - 0) = 2x \]. This tangent line shows how the curve behaves near \( a = 0 \).

Differentiation is a critical tool in calculus used to find the derivative of a function. It involves applying specific rules to functions to ascertain their derivatives, which then inform us of the functions' behavior.
For the task of finding the linearization of the function \( f(x) = x^2 + 2x \):

• We start by differentiating each term of the function separately using the sum and power rules.
• Side by side, we calculate: \( \frac{d}{dx}(x^2) = 2x \) and \( \frac{d}{dx}(2x) = 2 \).
• Combining these, we establish that \( f'(x) = 2x + 2 \), providing the expression for the slope at any point \( x \).

Differentiation simplifies complex curves into linear segments, making it easier to analyze and compute tangents and approximations.

In mathematical analysis, approximation refers to finding values close enough to exact numbers, often using simpler expressions or methods. Linearization serves as an approximation technique, offering an easy-to-handle line that mimics the behavior of a function near a given point.
For the function \( f(x) = x^2 + 2x \), the linearization at \( a = 0 \) gives us \( L(x) = 2x \). This is because at \( a = 0 \), the function itself is not complex (\( f(0) = 0 \)), and the slope derived from the derivative (\( f'(0) = 2 \)) helps in forming a straight line.
Approximations using linearization are especially helpful when:

• The actual function is difficult to compute for each desired x-value.
• A quick estimation is needed.

Approximating with linearization is a handy way to simplify analyses and calculations, capturing essential trends and behaviors of functions without overwhelming variety.