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

Problem

d(ln(ln(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(x)

2. Apply the Chain Rule which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Differentiate the outer function with respect to its argument, which gives 1/ln(x)

4. Differentiate the inner function ln(x) with respect to x which gives 1/x

5. Multiply the results of the derivatives together.

6. Simplify the expression by combining the terms into a single fraction.

Final Answer

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

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