Find the Derivative - d/dx ( natural log of x)/(x^6)

Problem

d()/d(x)ln(x)/(x^6)

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 numerator and denominator. Let u=ln(x) and v=x^6

3. Differentiate each part individually. The derivative of the numerator is d(ln(x))/d(x)=1/x and the derivative of the denominator is d(x^6)/d(x)=6*x^5

4. Substitute these components into the quotient rule formula.

d()/d(x)ln(x)/(x^6)=(x^6⋅1/x−ln(x)⋅6*x^5)/((x^6)^2)

5. Simplify the numerator by multiplying the terms.

d()/d(x)ln(x)/(x^6)=(x^5−6*x^5*ln(x))/(x^12)

6. Factor out the common term x^5 from the numerator to further simplify the fraction.

d()/d(x)ln(x)/(x^6)=(x^5*(1−6*ln(x)))/(x^12)

7. Reduce the expression by dividing both the numerator and denominator by x^5

d()/d(x)ln(x)/(x^6)=(1−6*ln(x))/(x^7)

Final Answer

d()/d(x)ln(x)/(x^6)=(1−6*ln(x))/(x^7)

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