Find the Derivative - d/dx y=( natural log of x)/(x^5)

Problem

d()/d(x)ln(x)/(x5)

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=x5

3. Differentiate the individual components: d(ln(x))/d(x)=1/x and d(x5)/d(x)=5*x4

4. Substitute these values into the quotient rule formula:

d()/d(x)ln(x)/(x5)=(x5⋅1/x−ln(x)⋅5*x4)/((x5)2)

5. Simplify the numerator by performing the multiplication:

d()/d(x)ln(x)/(x5)=(x4−5*x4*ln(x))/(x10)

6. Factor out the common term x4 from the numerator:

d()/d(x)ln(x)/(x5)=(x4*(1−5*ln(x)))/(x10)

7. Reduce the fraction by dividing both numerator and denominator by x4

d()/d(x)ln(x)/(x5)=(1−5*ln(x))/(x6)

Final Answer

d()/d(x)ln(x)/(x5)=(1−5*ln(x))/(x6)

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