Find the Derivative - d/dx ( natural log of x)/(4x)

Problem

d()/d(x)ln(x)/(4*x)

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))/(v2)

2. Assign the variables for the numerator and denominator. Let u=ln(x) and v=4*x

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(4)*x)/d(x)=4

4. Substitute these values into the quotient rule formula.

d()/d(x)ln(x)/(4*x)=(4*x⋅1/x−ln(x)⋅4)/((4*x)2)

5. Simplify the numerator by multiplying 4*x and 1/x to get 4

d()/d(x)ln(x)/(4*x)=(4−4*ln(x))/(16*x2)

6. Factor out the common factor of 4 in the numerator to simplify the fraction further.

d()/d(x)ln(x)/(4*x)=(4*(1−ln(x)))/(16*x2)

7. Reduce the fraction by dividing both the numerator and denominator by 4

d()/d(x)ln(x)/(4*x)=(1−ln(x))/(4*x2)

Final Answer

d()/d(x)ln(x)/(4*x)=(1−ln(x))/(4*x2)

Want more problems? Check here!