Find the Derivative - d/dx ( natural log of x)/(8x)

Problem

d()/d(x)ln(x)/(8*x)

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=8*x

3. Differentiate the individual components: d(ln(x))/d(x)=1/x and (d(8)*x)/d(x)=8

4. Substitute these values into the quotient rule formula:

d()/d(x)ln(x)/(8*x)=((8*x)*(1/x)−(ln(x))*(8))/((8*x)2)

5. Simplify the numerator by multiplying 8*x and 1/x

d()/d(x)ln(x)/(8*x)=(8−8*ln(x))/(64*x2)

6. Factor out the common factor of 8 in the numerator:

d()/d(x)ln(x)/(8*x)=(8*(1−ln(x)))/(64*x2)

7. Reduce the fraction by dividing both the numerator and denominator by 8:

d()/d(x)ln(x)/(8*x)=(1−ln(x))/(8*x2)

Final Answer

d()/d(x)ln(x)/(8*x)=(1−ln(x))/(8*x2)

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