91Ó°ÊÓ

The concept of a derivative is fundamental in calculus as it represents the rate at which a function changes. To find the derivative of a function, we differentiate it. For the function given,
\( f(x) = x^2 + 2x \),
we need to calculate its derivative to assist in forming a linear approximation. By using the power rule, which states that the derivative of
\( x^n \)
is
\( nx^{n-1} \),
we find the derivative to be
\( f'(x) = 2x + 2 \).
This derivative tells us how the function
\( f(x) \)
changes at any point
\( x \).
We use this rate of change to find linear approximations at points near
\( a \).

Function evaluation involves substituting a specific value into a function to determine its output. When we need to perform a linearization, evaluating both the function and its derivative at strategic points simplifies the process.
For instance, using
\( f(x) = x^2 + 2x \),
we evaluated this function at
\( x = 0 \),
which is close to
\( a = 0.1 \).
This gave us
\( f(0) = 0 \).
Similarly, the derivative
\( f'(x) = 2x + 2 \)
was evaluated at
\( x = 0 \)
resulting in
\( f'(0) = 2 \).
Evaluating these functions at easy to compute points helps directly in the construction of linear approximations, making calculations simpler and more efficient on paper and in practice.

Linear approximation is a technique used to approximate the value of a function near a point using information about its derivative.
This method is based on using the tangent line at a point to estimate the function’s output values nearby.
For the function given,
we linearized it at
\( x = 0 \),
where both the function
\( f(x) = x^2 + 2x \)
and its derivative were easy to compute. By substituting these into the linearization formula,

• \( L(x) = f(a) + f'(a)(x - a) \)

with
\( f(0) = 0 \) and
\( f'(0) = 2 \),
we derived the linear approximation:
\( L(x) = 2x \).
This approximation simplifies calculations for function values close to
\( x = 0.1 \),
and helps in understanding the function’s behavior around the point of interest.

Differentiation involves finding the derivative of a function, a process central to calculus that allows us to understand the function's behavior, such as rates of change and slopes of tangent lines.
For the function
\( f(x) = x^2 + 2x \),
the differentiation produces
\( f'(x) = 2x + 2 \).
This formula provides crucial information about the function, including where it increases or decreases.
When differentiating, it is crucial to apply the rules of differentiation accurately, such as the power rule, product rule, etc., depending on the complexity of the function. Differentiation enables us to move from there towards applications like linearization, where understanding the rate of change aids in constructing useful approximations or predictions about the function's behavior near specific points. This allows us to approximate values in a way that is both accurate and easy to calculate.