Find the Derivative - d/dx y=( natural log of x)/x

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two terms, ln(x) and x 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=x

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(x)/d(x)=1

4. Substitute these values into the quotient rule formula.

d(y)/d(x)=(x⋅1/x−ln(x)⋅1)/(x2)

5. Simplify the expression in the numerator. Since x⋅1/x=1 the expression becomes:

d(y)/d(x)=(1−ln(x))/(x2)

Final Answer

d()/d(x)ln(x)/x=(1−ln(x))/(x2)

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