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

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two terms, ln(x) and x2 use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables for the quotient rule where u=ln(x) and v=x2

3. Differentiate the individual components. The derivative of the numerator is d(ln(x))/d(x)=1/x and the derivative of the denominator is d(x2)/d(x)=2*x

4. Substitute these components into the quotient rule formula.

d()/d(x)ln(x)/(x2)=(x21/x−ln(x)*(2*x))/((x2)2)

5. Simplify the numerator by multiplying the terms.

d()/d(x)ln(x)/(x2)=(x−2*x*ln(x))/(x4)

6. Factor out a common factor of x from the numerator to further simplify the expression.

d()/d(x)ln(x)/(x2)=(x*(1−2*ln(x)))/(x4)

7. Cancel the common factor of x in the numerator and denominator.

d()/d(x)ln(x)/(x2)=(1−2*ln(x))/(x3)

Final Answer

d()/d(x)ln(x)/(x2)=(1−2*ln(x))/(x3)

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