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The product rule is a fundamental concept in calculus, used when taking the derivative of a product of two functions. Imagine you have two functions: one is \( u(x) \), and the other is \( v(x) \). When they are multiplied together to form a new function, \( f(x) = u(x) \cdot v(x) \), their derivative \( f'(x) \) is not simply the product of their derivatives. Instead, you use the product rule.The product rule states:

• \( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

This formula tells us how to handle the complexity of two changing quantities multiplying each other.Let's see what happens in our specific case:- We have \( f(x) = x \ln x \).- Here, \( u = x \) and \( v = \ln x \).- Their derivatives are \( u'(x) = 1 \) and \( v'(x) = \frac{1}{x} \) respectively.Plug these into the formula:

• \( f'(x) = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1 \)

This method allows us to accurately find the derivative even when functions are multiplied together. It's a handy tool to have in your calculus toolkit.

The power rule simplifies the process of differentiating functions that are powers of \( x \). It's a straightforward rule used in calculus which states:

• If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

This means you bring down the exponent as a coefficient and then decrease the exponent by 1. It's a quick way to find derivatives without complicated algebra.In our problem, the power rule is used when differentiating \( x \). Here, we see \( x \) raised to the first power:- When using the power rule, \( x \) becomes \( 1 \cdot x^{1-1} = 1 \cdot x^0 = 1 \).This gives us the derivative of \( u(x) = x \) as \( u'(x) = 1 \). It's simple but essential, particularly when combined with other rules like the product rule to handle more complex functions.

The natural logarithm is a particular type of logarithm with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. This logarithm is denoted as \( \ln x \) and has unique derivative properties that are useful in calculus.When differentiating \( \ln x \), the rule is straightforward:

• The derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \).

This means that for any function involving the natural logarithm, knowing this derivative rule is crucial. In this problem, we deal with a function \( v(x) = \ln x \). So the derivative \( v'(x) \) is found using the rule:- \( v'(x) = \frac{1}{x} \).This result is then used in the application of the product rule. The natural logarithm's properties are incredibly useful, particularly in problems involving exponential growth or decay. Understanding its derivative helps tackle a wide array of mathematical challenges easily.