Find dy/dx y=x natural log of x

Problem

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

Solution

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

2. Recall the product rule formula for differentiation: 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 into the product rule formula: d(y)/d(x)=x⋅1/x+ln(x)⋅1

5. Simplify the expression by canceling x in the first term and multiplying in the second term.

Final Answer

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

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