Find the Derivative - d/dx ( natural log of x)/(x^7)

Problem

d()/d(x)ln(x)/(x^7)

Solution

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

2. Assign the variables for the quotient rule where u=ln(x) and v=x^7

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^7)/d(x)=7*x^6

4. Substitute these values into the quotient rule formula.

d()/d(x)ln(x)/(x^7)=(x^7⋅1/x−ln(x)⋅7*x^6)/((x^7)^2)

5. Simplify the numerator by multiplying x^7 and 1/x to get x^6

d()/d(x)ln(x)/(x^7)=(x^6−7*x^6*ln(x))/(x^14)

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

d()/d(x)ln(x)/(x^7)=(x^6*(1−7*ln(x)))/(x^14)

7. Reduce the fraction by dividing both the numerator and denominator by x^6

d()/d(x)ln(x)/(x^7)=(1−7*ln(x))/(x^8)

Final Answer

d()/d(x)ln(x)/(x^7)=(1−7*ln(x))/(x^8)

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