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Differentiation is the process of finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point. It helps us understand how variables are linked and how they influence each other over a tiny interval. In simple terms, differentiation is like measuring how steep a hill is at a particular spot. When we differentiate a function, we are looking at how the value of the function changes as the input changes slightly.
To differentiate a function like the one in our problem, we apply specific rules of calculus, which is a branch of mathematics centered around change. This gives us a new function, called the derivative, that tells us the slope of the tangent line to the curve at any point.
Understanding differentiation allows us to solve many real-world problems, from physics and engineering to economics and biology, by predicting how a system will behave when small changes occur.

The power rule is a fundamental technique in calculus used to differentiate functions of the form \(x^n\), where \(n\) is any real number. It states that if \(y = x^n\), its derivative \(\frac{dy}{dx}\) is given by \(nx^{n-1}\). This rule simplifies the differentiation process significantly and is applicable to both positive and negative powers, as well as fractional exponents.
In our original exercise, the function involves the term \(-\frac{1}{x}\), which can be rewritten using exponents as \(-x^{-1}\). By applying the power rule, \(n = -1\), we find the derivative of this term to be \(x^{-2}\) or \(\frac{1}{x^2}\), because \(-1 \times x^{-2} = x^{-2}\). This showcases the utility of the power rule in tackling problems involving different types of exponents smoothly.
This powerful tool helps to quickly evaluate how functions behave and changes are involved, without the need for complex algebraic manipulations.

Once a derivative is determined, evaluating it at a specific point gives a tangible value that conveys how the function is changing exactly at that point. In our problem, after differentiating the function, we evaluated it at \(x = \sqrt{3}\). This step involves substituting \(x = \sqrt{3}\) into the derived expression \(\frac{1}{x^2}\) to find the rate of change at that specific location.
In practice, evaluating a derivative provides insights such as velocity of an object, gradient of a surface, or rate of profit growth. It converts the general formula obtained from differentiation into a specific, usable number.
For our function, by substituting \(x = \sqrt{3}\), we calculated \(\frac{1}{3}\), which shows that at \(x = \sqrt{3}\), the function is slowly increasing. This precise rate of change can be critical in applications such as predicting future values or making decisions based on instant slopes.