Find the Derivative - d/dx natural log of natural log of 2x

Problem

d(ln(ln(2*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(2*x)

2. Apply the Chain Rule for the natural logarithm, which states d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Differentiate the inner function u=ln(2*x) This requires another application of the Chain Rule where the new inner function is 2*x

4. Calculate the derivative of the innermost part (d(2)*x)/d(x)=2

5. Combine the results of the derivatives.

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

6. Substitute the derivative of the second natural log.

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

7. Simplify the expression by canceling the constant factor 2

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

Final Answer

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

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