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

Problem

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

Solution

1. Identify the rule needed for the derivative of a quotient, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions: u=ln(x) and v=x2

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

4. Substitute these values into the quotient rule formula.

(x2⋅1/x−ln(x)⋅2*x)/((x2)2)

5. Simplify the numerator by performing the multiplication x2⋅1/x=x

(x−2*x*ln(x))/(x4)

6. Factor out the common term x from the numerator.

(x*(1−2*ln(x)))/(x4)

7. Reduce the fraction by dividing both the numerator and denominator by x

(1−2*ln(x))/(x3)

Final Answer

d()/d(x)ln(x)/(x2)=(1−2*ln(x))/(x3)

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