Find the Derivative - d/dx natural log of natural log of 8x

Problem

d(ln(ln(8*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(8*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(8*x) by applying the chain rule again.

4. Calculate the derivative of the innermost part (d(8)*x)/d(x)=8

5. Combine the results of the chain rule steps.

d(ln(ln(8*x)))/d(x)=1/ln(8*x)⋅d(ln(8*x))/d(x)

6. Substitute the derivative of the inner natural log.

d(ln(ln(8*x)))/d(x)=1/ln(8*x)⋅1/(8*x)⋅8

7. Simplify the expression by canceling the constant factor 8.

d(ln(ln(8*x)))/d(x)=1/(x*ln(8*x))

Final Answer

d(ln(ln(8*x)))/d(x)=1/(x*ln(8*x))

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