Find the Derivative - d/dx x^2 natural log of x

Problem

d()/d(x)*x2*ln(x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=x2 and v(x)=ln(x)

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

3. Differentiate the individual components: d(x2)/d(x)=2*x and d(ln(x))/d(x)=1/x

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

x2⋅1/x+ln(x)⋅2*x

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

x+2*x*ln(x)

6. Factor out the common term x to reach the final form.

x*(1+2*ln(x))

Final Answer

d()/d(x)*x2*ln(x)=x*(1+2*ln(x))

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