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The process of finding a derivative involves calculating the rate at which a function changes at any given point. This is fundamental in calculus and serves as the groundwork for understanding changes in various scenarios. To find a derivative, you are essentially determining the slope or the steepness of the curve represented by the function.Differentiation is symbolized by the prime notation, like \( f'(x) \), or by using the Leibniz notation \( \frac{dy}{dx} \). The derivative tells us how much \( y \) changes as \( x \) changes. For example, given a function \( f(x) = \frac{x-1}{x} \), our first step is to differentiate it to find \( f'(x) \).Simplifying the function initially makes this process more straightforward. After simplifying \( f(x) = \frac{x-1}{x} \) to \( f(x) = 1 - \frac{1}{x} \), you find \( f'(x) = \frac{1}{x^2} \). This indicates how the function's value changes as \( x \) changes, reflecting how rapidly or slowly \( f(x) \) strengthens or weakens at each point.

Once we have the first derivative, the second derivative gives additional insight. It indicates the behavior and shape of the graph of a function. Specifically, the second derivative tells us about the curvature or concavity of the function.Calculating the second derivative involves differentiating the first derivative. For the function in our example, with the first derivative \( f'(x) = \frac{1}{x^2} \), we rewrite it as \( x^{-2} \). Differentiating this with respect to \( x \) yields the second derivative: \( f''(x) = -2x^{-3} = -\frac{2}{x^3} \).The second derivative thus informs us whether the curve is concave up or concave down at different points. If \( f''(x) > 0 \), the graph is concave up, while \( f''(x) < 0 \) indicates it's concave down. By evaluating \( f''(x) \) at particular points, such as \( x = 3 \), we obtain \( f''(3) = -\frac{2}{27} \), suggesting that near \( x=3 \), the function curves downward.

The power rule is a crucial tool in differentiation. It's an easy and effective method to find the derivative of functions that are simple polynomial expressions. The rule states: if you have a function \( f(x) = x^n \), its derivative \( f'(x) \) is \( nx^{n-1} \).This rule is particularly useful when dealing with functions where terms are individual powers of \( x \). In our problem, while simplifying and differentiating \( f(x) = 1 - \frac{1}{x} \), applying the power rule to \( -x^{-1} \) resulted in the derivative \( \,\frac{1}{x^2} \). Similarly, it's used again to find \( f''(x) \) from \( f'(x) = x^{-2} \), resulting in \( f''(x) = -2x^{-3} \).By using the power rule, the cumbersome task of differentiation becomes straightforward, as it allows quickly finding first, second, or even higher-order derivatives for polynomials. This understanding simplifies many tasks in calculus, making analysis quicker and more efficient.