Find the Derivative - d/dx y=( natural log of x)/(x^3)

Problem

d()/d(x)ln(x)/(x^3)

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two functions, u=ln(x) and v=x^3 apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Differentiate the numerator u=ln(x) to get d(u)/d(x)=1/x

3. Differentiate the denominator v=x^3 using the power rule to get d(v)/d(x)=3*x^2

4. Substitute these components into the quotient rule formula.

d()/d(x)ln(x)/(x^3)=(x^3⋅1/x−ln(x)⋅3*x^2)/((x^3)^2)

5. Simplify the numerator by multiplying x^3⋅1/x=x^2 and the denominator by applying the power of a power rule (x^3)^2=x^6

d()/d(x)ln(x)/(x^3)=(x^2−3*x^2*ln(x))/(x^6)

6. Factor out the common term x^2 from the numerator to simplify the fraction.

d()/d(x)ln(x)/(x^3)=(x^2*(1−3*ln(x)))/(x^6)

7. Divide both the numerator and denominator by x^2 to reach the final simplified form.

d()/d(x)ln(x)/(x^3)=(1−3*ln(x))/(x^4)

Final Answer

d()/d(x)ln(x)/(x^3)=(1−3*ln(x))/(x^4)

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