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

Verified step by step guidance

1. Identify the function for which you need to find the derivative: \( y = x \ln(x) - x \).
2. Apply the derivative rules: Use the product rule for \( x \ln(x) \) and the power rule for \( -x \).
3. For the term \( x \ln(x) \), apply the product rule: \( (u \cdot v)' = u'v + uv' \), where \( u = x \) and \( v = \ln(x) \).
4. Calculate the derivatives: \( u' = 1 \) and \( v' = \frac{1}{x} \). Substitute these into the product rule: \( (x \ln(x))' = 1 \cdot \ln(x) + x \cdot \frac{1}{x} \).
5. For the term \( -x \), apply the power rule: \( (x^n)' = nx^{n-1} \). Here, \( n = 1 \), so the derivative is \( -1 \). Combine the results to find the derivative of the entire function.

Key Concepts

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. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

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 products of simpler functions, such as the term x ln(x) in the given function.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately equal to 2.71828. It is a fundamental 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 differentiate functions involving natural logarithms, such as ln(x) in the given function.