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The product rule is a fundamental technique in calculus used for differentiating functions that are products of two or more terms. When you have a function like \( f(x) = u(x) \cdot v(x) \), where \( u(x) \) and \( v(x) \) are both functions of \( x \), the derivative is calculated using the formula:

• \( f'(x) = u'(x)v(x) + u(x)v'(x) \)

This means you differentiate each function separately, then multiply each derivative by the other original function and sum the results.
When applying the product rule, it's crucial to correctly identify and differentiate each component to ensure accurate results. In our example, \( u = x^4 \) and \( v = \ln x \), leading to derivatives \( u'(x) = 4x^3 \) and \( v'(x) = \frac{1}{x} \). Substituting into the product rule gives the result for the derivative of the entire product.

Logarithmic functions are functions of the form \( f(x) = \ln x \), where \( \ln x \) represents the natural logarithm of \( x \). The natural logarithm is the inverse of the exponential function \( e^x \). It has special properties that make it particularly useful in calculus:

• Its derivative is particularly simple: for \( f(x) = \ln x \), \( f'(x) = \frac{1}{x} \).
• Logarithms help simplify multiplication into addition, which is useful for various mathematical manipulations.

In the context of differentiation, the derivative \( \frac{1}{x} \) plays a key role, especially when combined with other types of functions. The ease of differentiating logarithmic functions often simplifies complex calculus problems. In our example, understanding that the derivative of \( \ln x \) is \( \frac{1}{x} \) allows you to efficiently apply the product rule when differentiating the function \( f(x) = x^4 \ln x \).

Polynomial functions are expressions that consist of variables raised to positive integer powers. For example, a simple polynomial function could be \( x^4 \). These functions are fundamental in calculus because of their simplicity and predictable patterns:

• The derivative of a polynomial function is defined by the power rule: if \( f(x) = ax^n \), then \( f'(x) = nax^{n-1} \).
• This rule makes differentiation straightforward because it allows you to systematically reduce the power of \( x \) by one and multiply by the original power.

Understanding polynomial functions is crucial for applying techniques like the product rule, especially when they form one part of a product with non-polynomial functions, such as logarithmic functions. In our exercise, \( u(x) = x^4 \) was differentiated using the power rule to get \( u'(x) = 4x^3 \), allowing you to compute \( f'(x) \) accurately when combined with the derivative of \( \ln x \).