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

Problem

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

Solution

1. Identify the rule needed for the derivative of a product of two functions, u=x2 and v=ln(2*x)

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

3. Differentiate the first part u=x2 to get d(u)/d(x)=2*x

4. Differentiate the second part v=ln(2*x) using the chain rule to get d(v)/d(x)=1/(2*x)⋅2=1/x

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

x2⋅1/x+ln(2*x)⋅2*x

6. Simplify the resulting expression by canceling terms and rearranging.

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

Final Answer

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

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