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

Problem

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

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is ln(u) where u=ln(5*x)

2. Apply the Chain Rule for the outer natural log. The derivative of ln(u) is 1/u⋅d(u)/d(x)

d(ln(ln(5*x)))/d(x)=1/ln(5*x)⋅d(ln(5*x))/d(x)

3. Apply the Chain Rule again for the inner natural log. The derivative of ln(5*x) is 1/(5*x)⋅(d(5)*x)/d(x)

d(ln(5*x))/d(x)=1/(5*x)⋅5

4. Simplify the derivative of the inner function by canceling the constant factor 5

1/(5*x)⋅5=1/x

5. Combine the results from the previous steps to find the final derivative.

1/ln(5*x)⋅1/x=1/(x*ln(5*x))

Final Answer

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

Want more problems? Check here!