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The product rule is a fundamental technique used to differentiate functions that are products of two or more simpler functions. This rule simplifies the process of finding derivatives of composite functions. It states that if you have a product of two functions, say \( u(x) \) and \( v(x) \), the derivative of their product is given by:

• \[ (uv)' = u'v + uv' \]

This formula allows us to break down the task into finding the derivatives of the individual components \( u(x) \) and \( v(x) \), and then applying them together in the formula above.
In the example exercise, the term \( x \ln x \) is such a product where \( u = x \) and \( v = \ln x \). Here,

• \( u' = 1 \) because the derivative of \( x \) is \( 1 \).
• \( v' = \frac{1}{x} \) because the derivative of \( \ln x \) is \( \frac{1}{x} \).

By applying the product rule formula, we can find the derivative of \( x \ln x \) as: \( (x \ln x)' = \ln x + 1 \). This method is essential because it allows us to handle more complex functions found in calculus.

Logarithmic differentiation is a powerful technique, especially useful when the function involves products, quotients, or powers of functions. It involves taking the natural logarithm of both sides of an equation and then differentiating using the properties of logarithms.
Using logarithmic differentiation can simplify the process by turning multiplicative processes into additive, thanks to the rule that \( \ln(ab) = \ln a + \ln b \). While it wasn’t directly used in this solution, it is an important technique to understand when dealing with complex functions involving logarithms.
In cases where the function is in the form of \( x^x \) or similar, logarithmic differentiation makes things easier by simplifying exponentials and products for easier differentiation. To apply, set \( y = f(x) \) and take the natural log on both sides so it becomes \( \ln y = \ln(f(x)) \). Differentiate both sides, using the chain rule on \( y \), to solve for \( y' \).
This approach is often used in conjunction with other rules, like the product rule, for maximum effectiveness.

Differentiation techniques in calculus are essential for finding the rate of change of functions. They provide a range of methods for handling different forms and complexities of functions. Common techniques include:

• Power Rule: Easily finds the derivative of any power of \( x \), \( \frac{d}{dx}(x^n) = nx^{n-1} \).
• Product Rule: As previously discussed, helpful for differentiating products of functions.
• Quotient Rule: Useful for differentiating quotients, \( \frac{u}{v} \), where \( (\frac{u}{v})' = \frac{u'v - uv'}{v^2} \).
• Chain Rule: Necessary when differentiating composite functions, allowing us to work from the outermost to the innermost function layers.

Each technique has a specific use-case that makes it effective under certain conditions. By understanding and recognizing which rule to apply, finding the derivatives of more intricate functions becomes manageable.
In our exercise, the "simpler" rules like the product rule were used to break down the work into manageable steps, demonstrating their value in practical applications.