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The difference quotient is a technique used to estimate the derivative of a function at a particular point. It involves evaluating the function at a point slightly offset by a small increment, often denoted as \(h\). This method is based on the definition of the derivative as \(f'(x) \approx \frac{f(x + h) - f(x)}{h}\).

• When \(h\) is very small, the difference quotient provides a good approximation of the derivative.
• This approach is especially useful for estimating derivatives when algebraic differentiation is complicated or impractical.

The example function \(f(x) = 2x + x^{-1}\) illustrates this technique by estimating \(f'(1)\) using small values of \(h\) like 0.1, 0.01, or 0.001. By substituting these tiny increments into the difference quotient, we get a close approximation for \(f'(1)\). However, this method can lead to precision issues for extremely small \(h\) due to limitations in numerical computation. To address these issues, algebraic differentiation can be used to find the derivative exactly.

Algebraic differentiation is the process of finding the exact derivative of a function using algebraic techniques. Unlike the estimation given by the difference quotient, algebraic differentiation provides an exact symbolic result.

• It is often performed by applying rules for differentiation to find a new function that represents the rate of change of the original function.
• This method eliminates the approximation error associated with numerical methods, making it highly accurate and preferable when an exact value is required.

In the given exercise, we use algebraic differentiation to differentiate \(f(x) = 2x + x^{-1}\). This involves differentiating each component of the function:
- The derivative of \(2x\) is 2 because the slope of a line described by \(2x\) is constant.- The derivative of \(x^{-1}\) requires the power rule, yielding \(-x^{-2}\).
Thus, the exact derivative is \(f'(x) = 2 - x^{-2}\), which gives us the precise rate of change at any point \(x\), including \(x = 1\), leading to \(f'(1) = 1\).

The power rule is a fundamental principle in calculus used for differentiating functions where variables are raised to a power. This rule makes deriving any polynomial function straightforward.
In mathematical terms, if you have a function in the form \(x^n\), its derivative using the power rule is \((x^n)' = nx^{n-1}\). This indicates that you multiply the variable's exponent by its coefficient and then reduce the exponent by one.

• This rule is efficient and widely applicable across various problems involving polynomials.
• It simplifies the process of differentiation, converting a potentially complex task into an easy procedure.

For the exercise function \(f(x) = 2x + x^{-1}\), the term \(x^{-1}\) can be treated as \(x^n\) where \(n = -1\). Applying the power rule gives the derivative of \(x^{-1}\) as \(-x^{-2}\). Employing the power rule effectively in algebraic differentiation provides an exact and reliable result, ensuring accuracy compared to approximation methods.