Find the Derivative - d/dx f(x)=x^2 natural log of x

Problem

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

Solution

1. Identify the rule needed for the expression x2*ln(x) which is the product of two functions u=x2 and v=ln(x)

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

x2d(ln(x))/d(x)+ln(x)d(x2)/d(x)

5. Simplify the resulting expression by performing the multiplication and combining terms.

x2⋅1/x+ln(x)⋅2*x

x+2*x*ln(x)

Final Answer

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

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