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The power rule is a fundamental technique in calculus for finding the derivative of power functions. It is particularly useful for functions in the form of \( y = x^n \), where \( n \) is any real number. According to the power rule, the derivative of \( x^n \) with respect to \( x \) is given by the formula:\[ \frac{d}{dx}x^n = nx^{n-1} \]This means you multiply the power \( n \) by the term \( x^{n-1} \). Let's break it down further:- Bring down the power as a coefficient.- Reduce the power by one.For example, if \( y = x^{3} \), applying the power rule gives \( \frac{dy}{dx} = 3x^{2} \). This procedure is quick and handy for simplifying and differentiating polynomial expressions. When applied to \( y = x^{1/2} \), as in our original problem, we get the derivative \( \frac{dy}{dx} = \frac{1}{2}x^{-1/2} \). Remember, understanding this rule helps you tackle many types of derivative problems efficiently.

The derivative of a square root function can be initially approached by rewriting it in a form where the power rule can be applied. The function \( y = \sqrt{x} \) can be rewritten as \( y = x^{1/2} \). This step transforms the square root into a power expression, enabling us to use the powerful tool of calculus, the power rule.1. Convert square roots into exponent form. - Replace \( \sqrt{x} \) with \( x^{1/2} \). 2. Apply the power rule to find the derivative. - The power rule states: \( \frac{d}{dx}x^n = nx^{n-1} \). - For \( y = x^{1/2} \), we calculate: \( \frac{dy}{dx} = \frac{1}{2}x^{-1/2} \).By understanding how to work with square roots, and expressing them as exponents, you open up a straightforward path to finding derivatives using familiar rules.

Simplifying expressions in calculus, especially derivatives, is an essential step to reach a more manageable and easy-to-understand form. Starting from where the derivative was found as \( \frac{dy}{dx} = \frac{1}{2}x^{-1/2} \), simplification involves expressing it in terms of positive exponents or more easily interpretable forms.Here's how we simplify:- Recognize that negative exponents indicate reciprocals.- Rewrite \( x^{-1/2} \) as \( \frac{1}{\sqrt{x}} \).So, this expression becomes:\[ \frac{dy}{dx} = \frac{1}{2} \times \frac{1}{\sqrt{x}} = \frac{1}{2\sqrt{x}} \]This simplification highlights the importance of transforming derivatives into forms that are easier to integrate into further problems or applications. Mastering how to manipulate and simplify mathematical expressions is crucial for success in calculus.