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

Problem

d()/d(x)ln(5*x)/(5*x)

Solution

1. Identify the rule needed for differentiation, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables where u=ln(5*x) and v=5*x

3. Differentiate the numerator u using the chain rule: d(ln(5*x))/d(x)=1/(5*x)⋅5=1/x

4. Differentiate the denominator v (d(5)*x)/d(x)=5

5. Substitute these components into the quotient rule formula: ((5*x)*(1/x)−(ln(5*x))*(5))/((5*x)2)

6. Simplify the numerator by multiplying terms: 5−5*ln(5*x)

7. Simplify the denominator by squaring the term: 25*x2

8. Factor out the common factor of 5 in the numerator and divide: (5*(1−ln(5*x)))/(25*x2)=(1−ln(5*x))/(5*x2)

Final Answer

d()/d(x)ln(5*x)/(5*x)=(1−ln(5*x))/(5*x2)

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