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

Problem

(d(x)*ln(x))/d(x)

Solution

1. Identify the function as a product of two terms, u=x and v=ln(x) which requires the product rule.

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

3. Differentiate the individual components: d(x)/d(x)=1 and d(ln(x))/d(x)=1/x

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

x⋅1/x+ln(x)⋅1

5. Simplify the resulting expression by canceling x in the first term.

1+ln(x)

Final Answer

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

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