Find the Derivative - d/dx natural log of natural log of 10x

Problem

d(ln(ln(10*x)))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule, where the outer function is ln(u) and u=ln(10*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 ln(10*x) using the Chain Rule again, where the derivative of ln(a*x) is 1/(a*x)⋅a

4. Simplify the expression by canceling the constant factor 10 in the numerator and denominator.

5. Combine the terms into a single fraction.

Final Answer

d(ln(ln(10*x)))/d(x)=1/(x*ln(10*x))

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