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The derivative of a function is a fundamental concept in calculus that measures how a function changes as its input changes. It describes the rate of change or the slope of the function at any point. For the given function, \( f(x) = \frac{1}{x} \), the derivative helps us find the slope of the tangent line at a specific point. To compute this, we apply the power rule, which is especially handy for functions like \( \frac{1}{x} \).

To use the power rule, rewrite \( f(x) \) as \( x^{-1} \). The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Therefore, the derivative \( f'(x) \) of our function is:

• Apply the exponent: \( -1 \times x^{-2} = -\frac{1}{x^2} \).

At \( x = 1 \), evaluating \( f'(x) \) gives us the slope of the tangent line, which is \( -1 \). Calculating derivatives is crucial for understanding how functions behave, especially when approximating them near certain points.

The point-slope form provides a straightforward way to write the equation of a line when you know a point on the line and the slope. This form is a staple in algebra and calculus for equations of lines. In general, the point-slope form is written as:

• \( y - y_1 = m(x - x_1) \),

where \( m \) is the slope and \( (x_1, y_1) \) is the given point on the line.

For the function \( f(x) = \frac{1}{x} \), we previously calculated that at \( x = 1 \), the slope \( m \) is \( -1 \) and the point of tangency is \( (1, 1) \). Substituting these into the point-slope form, the equation becomes:

• \( y - 1 = -1(x - 1) \).

This simple rearrangement helps in finding the exact line equation tangent to the curve at a specific point, making it easy to understand changes in the system.

Linear approximation, also known as the tangent line approximation, is a method used to approximate the value of a function at a given point using the tangent line. It provides a simple way to make predictions about a function's value when it's difficult to compute the actual value of a function directly.

The approximation works best when you're close to the point of tangency. In the case of \( f(x) = \frac{1}{x} \) at \( x = 1 \), the tangent line approximation is described by the equation \( y = -x + 2 \). This linear function serves to estimate the values of \( f(x) \) near \( x = 1 \).

Linear approximation essentially replaces the curved behavior of a function with the straight line behavior of the tangent line. This is particularly useful:

• When dealing with small Δx values, where the function doesn't deviate much from the tangent line.
• In contexts where computational simplicity is critical, and linear equations offer that simplicity.

Understanding linear approximation is fundamental because it extends the concept of tangent lines beyond theoretical exercises and into practical applications for estimations and predictions.