How do you find the derivative of # (ln(ln(ln(x))) #?

Apply function of function rule.

ln x is differentiable for x > 0.

ln(ln(ln(x))) is differentiable for

ln(ln(x)) > 0, meaning ln(x) > 1, and so, x > e..

To find the derivative of ( \ln(\ln(\ln(x))) ), you would use the chain rule. The chain rule states that if you have a composition of functions, you differentiate the outer function and then multiply it by the derivative of the inner function.

Here's the step-by-step process:

1. Let ( u = \ln(\ln(x)) ).
2. Let ( v = \ln(u) ).
3. Let ( w = \ln(v) = \ln(\ln(\ln(x))) ).
4. Now, differentiate each function with respect to its variable, starting from the inside:
   • ( \frac{dw}{dv} = \frac{1}{v} )
   • ( \frac{dv}{du} = \frac{1}{u} )
   • ( \frac{du}{dx} = \frac{1}{x} \cdot \frac{1}{\ln(x)} )
5. Apply the chain rule:
   • ( \frac{dw}{dx} = \frac{dw}{dv} \cdot \frac{dv}{du} \cdot \frac{du}{dx} )
6. Substitute the derivatives:
   • ( \frac{dw}{dx} = \frac{1}{v} \cdot \frac{1}{u} \cdot \frac{1}{x \ln(x)} )
7. Substitute back the expressions for ( u ) and ( v ):
   • ( \frac{dw}{dx} = \frac{1}{\ln(\ln(x))} \cdot \frac{1}{\ln(x)} \cdot \frac{1}{x \ln(x)} )
8. Simplify:
   • ( \frac{dw}{dx} = \frac{1}{x \ln(x) \ln(\ln(x))} )

So, the derivative of ( \ln(\ln(\ln(x))) ) is ( \frac{1}{x \ln(x) \ln(\ln(x))} ).