Find the Derivative - d/dx x natural log of x-x

Problem

d()/d(x)*(x*ln(x)−x)

Solution

1. Identify the structure of the expression as a difference of two terms, x*ln(x) and x and apply the sum/difference rule for derivatives.

d()/d(x)*(x*ln(x)−x)=(d(x)*ln(x))/d(x)−d(x)/d(x)

2. Apply the product rule to the first term, x*ln(x) which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

(d(x)*ln(x))/d(x)=x⋅d(ln(x))/d(x)+ln(x)⋅d(x)/d(x)

3. Differentiate the individual components using the power rule and the derivative of the natural logarithm.

d(ln(x))/d(x)=1/x

d(x)/d(x)=1

4. Substitute these derivatives back into the product rule expression.

x⋅1/x+ln(x)⋅1

5. Simplify the resulting expression.

1+ln(x)

6. Combine all parts by subtracting the derivative of the second term of the original expression, which is d(x)/d(x)=1

(1+ln(x))−1

7. Finalize the simplification by canceling the constants.

ln(x)

Final Answer

d()/d(x)*(x*ln(x)−x)=ln(x)

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