Find the Derivative - d/dx natural log of natural log of 6x

Problem

d(ln(ln(6*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(6*x)

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

d(ln(ln(6*x)))/d(x)=1/ln(6*x)⋅d(ln(6*x))/d(x)

3. Differentiate the inner function ln(6*x) by applying the Chain Rule again.

d(ln(6*x))/d(x)=1/(6*x)⋅(d(6)*x)/d(x)

4. Calculate the derivative of 6*x which is 6

d(ln(6*x))/d(x)=1/(6*x)⋅6

5. Simplify the derivative of the inner function.

d(ln(6*x))/d(x)=1/x

6. Substitute the result back into the main expression.

d(ln(ln(6*x)))/d(x)=1/ln(6*x)⋅1/x

7. Combine the terms into a single fraction.

d(ln(ln(6*x)))/d(x)=1/(x*ln(6*x))

Final Answer

d(ln(ln(6*x)))/d(x)=1/(x*ln(6*x))

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