Find the Derivative - d/dx f(x)=x natural log of x-x

Problem

d()/d(x)*(x*ln(x)−x)

Solution

1. Identify the structure of the function, which consists of a product x*ln(x) and a linear term −x

2. Apply the product rule to the first term x*ln(x) where d()/d(x)*[u*v]=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part of the product rule:

xd(ln(x))/d(x)=x⋅1/x=1

4. Differentiate the second part of the product rule:

ln(x)d(x)/d(x)=ln(x)⋅1=ln(x)

5. Differentiate the remaining term −x

d(−x)/d(x)=−1

6. Combine all the results to find the derivative of the entire expression:

1+ln(x)−1=ln(x)

Final Answer

d()/d(x)*(x*ln(x)−x)=ln(x)

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