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The **Power Rule** is a fundamental concept in calculus that simplifies the process of differentiation. If you are not familiar with differentiation, think of it as finding how a function changes, or its slope, at any given point. The Power Rule specifically helps in finding the derivative of a term that is powered by an exponent.

When using the Power Rule, remember this formula: if you have a function of the form \( x^n \), its derivative is \( nx^{n-1} \). This means you multiply the current exponent by the coefficient and then reduce the exponent by one.

• Example: For \( x^{3/2} \), you multiply with \( \frac{3}{2} \), making it \( \frac{3}{2}x^{3/2-1} \) or \( \frac{3}{2}x^{1/2} \).
• Another example: For \( -6x^{-1} \), applying the Power Rule gives \( -6 \times (-1)x^{-1-1} = 6x^{-2} \).

The Power Rule is incredibly handy because it works with both positive and negative exponents, making it versatile for various mathematical expressions.

Before differentiating a function, especially when dealing with roots or fractions, it's crucial to express them as exponents. This is known as **rewriting with exponents**. By simplifying expressions in this way, applying the Power Rule becomes straightforward and eliminates potential confusion.

Let's take a closer look at how to perform this:

• **Square roots**: The square root of \( x \), written as \( \sqrt{x} \), can be rewritten as \( x^{1/2} \). This representation is more manageable for differentiation.
• **Fractional powers**: In expressions where there are roots multiplied by variables, such as \( x\sqrt{x} \), convert it to exponential form: \( x \times x^{1/2} = x^{3/2} \).
• **Negative exponents**: Fractions like \( \frac{1}{x} \) can be rewritten as \( x^{-1} \). Multiplying with constants remains unchanged, so \( \frac{6}{x} = 6x^{-1} \).

This rewriting is a preliminary step that prepares a function for smooth and efficient use of the Power Rule.

**Function Differentiation** is the core of understanding how functions behave by calculating their derivative. It shows the rate at which a function is changing at any point, represented by the slope of the tangent line to a curve.

In the function provided, \( y = x\sqrt{x} - \frac{6}{x} \), differentiation breaks down into applying rules systematically:

• First, rewrite and separate components if necessary.
• Apply the Power Rule to each term, like \( x^{3/2} \) and \(-6x^{-1}\), to find each derivative individually.
• Combine the derived terms to get a comprehensive derivative, which in this case results in \( y' = \frac{3}{2}x^{1/2} + 6x^{-2} \).

Each derivative provides a glimpse into how different components of the function influence its overall behavior. When combined, these components offer a clear picture of how the original function changes.