91Ó°ÊÓ

Simplifying a function before differentiating can make the differentiation process much easier and clearer. In the given problem, the function is initially presented as a complex fraction:
\[ W(x)=\frac{x^{2}+2 x-3+4 x \cos x}{3 x} \]
To simplify, we break it down into individual fractions:
\[ W(x)=\frac{x^2}{3x}+\frac{2x}{3x}-\frac{3}{3x}+\frac{4x \cos x}{3x} \]
By simplifying each term separately, we get:
\[ \frac{x^2}{3x}= \frac{x}{3}, \frac{2x}{3x}=\frac{2}{3}, -\frac{3}{3x}= -\frac{1}{x}, \frac{4x \cos x}{3x}= \frac{4 \cos x}{3} \]
Thus, the simplified function is
\[ W(x)= \frac{x}{3}+\frac{2}{3}-\frac{1}{x}+\frac{4 \cos x}{3} \]
This simplification makes it far easier to apply the differentiation rules.

Differentiation rules are formulas that help us find derivatives of various types of functions. Key rules used in this problem include:

• Constant Rule: The derivative of a constant is 0. For example,\[ \frac{d}{dx}(\frac{2}{3}) = 0 \]
• Power Rule: The derivative of \(x^n\) is \(nx^{n-1}\). For example,\[ \frac{d}{dx}( \frac{x}{3})= \frac{1}{3} \]
• Reciprocal Rule: The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \). In this problem, \[ \frac{d}{dx}(- \frac{1}{x}) = \frac{1}{x^2} \]
• Product Rule: Used when differentiating products of functions, which simplifies our work with complex terms.

Simplified differentiation can transform an initially challenging task into a manageable series of steps.

The chain rule is a fundamental differentiation rule used when differentiating compositions of functions. It states: \[ (f(g(x)))'= f'(g(x))g'(x) \]
In the given problem, we apply the chain rule to
\[ \frac{4 \cos x}{3} \]
The derivative of \( \cos x \) is \( -\sin x \). Thus,
\[ \frac{d}{dx}(\frac{4 \cos x}{3}) = \frac{4}{3} (-\sin x)= -\frac{4 \sin x}{3} \]
The chain rule is essential when dealing with functions nested within other functions.

Fractional derivatives are derivatives of terms with negative or fractional exponents. Here, we deal with terms like

• \(-\frac{1}{x}\), which can be viewed as \(x^{-1}\).

The derivative of \(x^{-1}\) using the power rule is \(-x^{-2}\), giving:
\[ \frac{d}{dx}( -\frac{1}{x})= x^{-1} \rightarrow- x^{-2} = \frac{1}{x^2} \]
Fractional derivatives are powerful tools for differentiating terms with fractional powers or roots. They follow the same rules but require careful handling of negative or fractional exponents.