Find the derivative of the following functions.
y = x² In x

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Identify the function to differentiate: \( y = x^2 \ln(x) \). This is a product of two functions: \( u = x^2 \) and \( v = \ln(x) \).

Apply the product rule for differentiation, which states that if \( y = u \cdot v \), then \( \frac{dy}{dx} = u'v + uv' \).

Differentiate \( u = x^2 \) with respect to \( x \). The derivative \( u' = \frac{d}{dx}(x^2) = 2x \).

Differentiate \( v = \ln(x) \) with respect to \( x \). The derivative \( v' = \frac{d}{dx}(\ln(x)) = \frac{1}{x} \).

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the product rule formula: \( \frac{dy}{dx} = (2x)(\ln(x)) + (x^2)(\frac{1}{x}) \). Simplify the expression to find the derivative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are used in various applications, including optimization and motion analysis.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as in the case of y = x² In x.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is an important function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, which is crucial when applying the product rule to functions that involve natural logarithms, such as In x in the given function.

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