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The power rule is a fundamental technique in calculus used to find the derivative of a function that is in the form of a power of x. It's essential for simplifying derivatives and is commonly used in many types of problems. The power rule states that if you have a term \(x^n\), the derivative is given by: \[ \frac{d}{dx}(x^n) = nx^{n-1} \] In other words, you multiply by the current power of x, then decrease the power by one. For example, consider the terms in our problem: \(x^{1/4}\) and \(-3x^{-1}\). Using the power rule for \(x^{1/4}\):

• The current power of x is 1/4.
• Multiply by 1/4.
• Decrease 1/4 by 1 to get \(1/4 - 1 = -3/4\).

This results in \(\frac{1}{4}x^{-3/4}\). Similarly, for \(-3x^{-1}\):

• The current power of x is -1.
• Multiply by -1.
• Decrease -1 by 1 to get \(-1 - 1 = -2\).

So, the derivative is \(3x^{-2}\).

Simplification is crucial for making a function easier to differentiate. In calculus, rewriting the function in more manageable terms can often turn a complex problem into a straightforward calculation. For our function, we start with: \[ \sqrt[4]{x} - \frac{3}{x} \] This expression is not readily in a form where we can directly apply the power rule. However, if we rewrite it, we get: \[ \frac{d}{dx}(x^{1/4} - 3x^{-1}) \] When we use the properties of exponents, we turn the radical \(\sqrt[4]{x}\) into \(x^{1/4}\) and the fraction \(-\frac{3}{x}\) into \(-3x^{-1}\). This makes it much easier to apply differentiation rules. Simplification methods like these often involve:

• Converting roots into fractional exponents.
• Converting fractions into negative exponents.

Differentiation involves following specific steps to systematically find the derivative. Let's break down the differentiation process for our problem: 1. **Simplify the Function:** Start by rewriting the function. We converted \(\sqrt[4]{x}\) to \(x^{1/4}\) and \(-\frac{3}{x}\) to \(-3x^{-1}\). 2. **Apply the Power Rule:** Differentiate each term separately. For \(x^{1/4}\), applying the power rule gives \(\frac{1}{4}x^{-3/4}\). For \(-3x^{-1}\), applying the power rule gives \(3x^{-2}\). 3. **Combine the Results:** After differentiating each term, combine them to get the final derivative: \[ \frac{1}{4}x^{-3/4} + 3x^{-2} \] Following these clearly defined steps helps break down complex calculus problems into manageable parts. Always remember, practice is key to mastering differentiation!