Derivative and second derivative

Question

I've got some homework and having a problem to solve some of them. Here are 3 of them where I need help, I need to differentiate them but no idea how I should do that.

1. Derivative of $e^{e^{e^{x}}}$

2. Derivative of $\ln(\ln(\ln x))$

3. Second derivative of $\ln(\ln x)$

Answer 1

the first derivative of #2 is given by $$f'(x)={\frac {1}{x\ln \left( x \right) \ln \left( \ln \left( x \right) \right) }} $$ using the chain rule and for #3 we get $$f'(x)={\frac {1}{x\ln \left( x \right) }}$$ and using the chanin rule and Quotient rule again: $$f''(x)=-{\frac {1}{{x}^{2}\ln \left( x \right) }}-{\frac {1}{{x}^{2} \left( \ln \left( x \right) \right) ^{2}}} $$ and check your answers

Answer 2

For the derivative of $f(x) = e^{e^{e^x}}$ you get, using the chain rule:

$f'(x) = e^{e^{e^x} + e^x + x}$.