91Ó°ÊÓ

Understanding how to calculate derivatives is a foundational skill in calculus. The derivative represents the rate at which a function's output changes with respect to a change in its input. Essentially, it's like the slope of the tangent line at any point on a graph of the function.

When we computed the derivative of the function \(f(x) = \frac{1}{x}\), what we were really finding out is how the function value changes as we make tiny movements along the x-axis. In this case, the derivative \(f'(x)\) tells us that for every unit we move away from a specific point \(a\), the function is changing at a rate proportional to \(\frac{1}{x^2}\).

This information is crucial when we want to estimate function values without calculating them exactly, which is the essence of the linear approximation method we employed to estimate \(\frac{1}{203}\).

The power rule for differentiation is an easy yet powerful tool to find the derivative of functions of the form \(x^n\), where \(n\) is any real number. The rule simply states that the derivative of \(x^n\) is \(nx^{n-1}\).

For our exercise, \(f(x) = \frac{1}{x} = x^{-1}\), so applying the power rule gives us \(f'(x) = -1\cdot x^{-1-1} = -\frac{1}{x^2}\). This straightforward technique bypasses the need for complex derivative calculations, thus simplifying our work significantly.

The accuracy of derivatives obtained using the power rule is indispensable for making reliable linear approximations, as it directly affects the slope of the line that approximates the function near a given point.

The linear approximation formula is a way to estimate the value of a function near a point \(a\) using its derivative. It states that \(f(x) \approx f(a) + f'(a) \cdot (x - a)\), where \(f'(a)\) is the derivative of \(f\) at \(a\). This is essentially the equation of the tangent line at \(a\), which we use as an estimate of \(f\) near \(a\).

In the given exercise, by using \(a = 200\), which is close to \(203\), we utilized this formula to find an approximated value of \(\frac{1}{203}\) based on the known value of \(\frac{1}{200}\). By adding the change (the product of the slope \(f'(a)\) and the distance \(x - a\)), we effectively 'walk' along the tangent line from \(a\) to our point of interest, \(203\), achieving an estimate close to the real value.

This powerful method is particularly useful when dealing with functions that are complex to evaluate directly but can be approximated near points where the function's value is known or easy to calculate.