Find the derivative of the following functions.
y = x² (1 - In x²)

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First, identify the function y = x² (1 - ln(x²)). This is a product of two functions: u(x) = x² and v(x) = 1 - ln(x²).

To find the derivative of y, apply the product rule: (u*v)' = u'v + uv'.

Calculate u'(x), the derivative of u(x) = x². Using the power rule, u'(x) = 2x.

Calculate v'(x), the derivative of v(x) = 1 - ln(x²). Use the chain rule: v'(x) = -d/dx[ln(x²)] = -2/x.

Substitute u'(x), v(x), u(x), and v'(x) into the product rule formula: y' = (2x)(1 - ln(x²)) + (x²)(-2/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 a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, as seen in the given function y = x²(1 - ln x²).

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately equal to 2.71828. It is a key function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, and understanding how to differentiate functions involving natural logarithms is crucial for solving problems that include ln terms, such as ln(x²) in the given function.

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