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

Problem

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

Solution

1. Apply the quotient rule for differentiation, which states that d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Identify the components where u=ln(x2) and v=x

3. Differentiate the numerator using the chain rule, where d(ln(x2))/d(x)=1/(x2)⋅2*x=2/x

4. Differentiate the denominator where d(x)/d(x)=1

5. Substitute the derivatives back into the quotient rule formula:

(x⋅2/x−ln(x2)⋅1)/(x2)

6. Simplify the expression by canceling x in the first term of the numerator.

(2−ln(x2))/(x2)

Final Answer

d()/d(x)ln(x2)/x=(2−ln(x2))/(x2)

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