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The product rule is a fundamental concept in calculus used when differentiating a product of two functions. If you have two differentiable functions, say \( u(x) \) and \( v(x) \), the product rule provides a structured way to compute the derivative of their product. The formula is given by:

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

This means to find the derivative of the product of two functions, you differentiate each separately and then add two terms:

• The first term is the derivative of the first function multiplied by the second function.
• The second term is the first function multiplied by the derivative of the second function.

For example, in the given problem where \( y = (x^2 + 1) \ln x \), applying the product rule involves:

• Identifying \( u(x) = x^2 + 1\) and \( v(x) = \ln x \).
• Computing \( u'(x) = 2x\) and \( v'(x) = \frac{1}{x}\).
• Substituting into the product rule formula gives \((2x)(\ln x) + (x^2 + 1)\left(\frac{1}{x}\right) \).

Differentiation is the process of finding the derivative of a function. This process helps determine the rate at which a function changes and is a vital tool in calculus for analyzing functions. In the context of the given exercise, we focus on differentiating polynomial and logarithmic functions, commonly encountered in calculus.

• The derivative of a polynomial function like \(x^2 + 1\) is calculated using simple rules where powers of \(x\) are multiplied by their respective exponents, resulting in \( 2x\).
• For logarithmic functions, such as \(\ln x\), the derivative is a standard result: \(\frac{1}{x}\).

Differentiation lets us explore how functions behave and change. Understanding these basics aids in tackling more complex calculus problems involving various rules and combinations of functions, such as the product rule encountered in this exercise.

Logarithmic functions are essential in calculus and come with unique differentiation rules. The most common is the natural logarithm function, denoted as \( \ln x \). Its derivative, \( \frac{d}{dx}(\ln x) = \frac{1}{x} \), is fundamental and frequently used in calculus problems.

• Logarithmic functions are defined for positive real numbers and are useful for expressing complex relationships in simpler terms.
• Their derivatives often appear in rules like the product rule or quotient rule due to their simplifying properties.

In the given problem, \(y = (x^2 + 1) \ln x\), the \(\ln x\) term crucially contributes to the overall differentiation process by making use of its straightforward derivative. It exemplifies how logarithmic functions simplify the work involved in analyzing the rate of change in compositions of functions. Understanding logarithmic functions and their properties is key to mastering calculus and seeing how they integrate into wider mathematical problems.