What is the derivative of the function xln(x) - x + C?

To find the derivative of the function ( f(x) = x \ln(x) - x + C ), we need to apply the rules of differentiation. This function consists of three parts: ( x \ln(x) ), ( -x ), and the constant ( C ).

First, we differentiate each part separately:

1. Derivative of ( x \ln(x) ): Using the product rule, which states that if you have two functions ( u ) and ( v ), then ( (uv)' = u'v + uv' ). Here, let ( u = x ) and ( v = \ln(x) ). The derivatives are ( u' = 1 ) and ( v' = \frac{1}{x} ). Thus, the derivative of ( x \ln(x) ) is calculated as:

[

(x \ln(x))' = 1 \cdot \ln(x) + x \cdot \frac{1}{x} = \ln(x) + 1.

]

2. Derivative of ( -x ): The derivative of ( -x ) is simply ( -1 ).

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