Find the derivative of $LaTeX: \displaystyle f(x) = 5^{5^{5^{x}}}$

Taking the derivative with the chain rule gives $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$ . $LaTeX: \displaystyle f'(x) = (5^{u} \ln{\left(5 \right)})(5^{v} \ln{\left(5 \right)})(5^{x} \ln{\left(5 \right)})$ . Substituting back in $LaTeX: \displaystyle u$ and $LaTeX: \displaystyle v$ gives $LaTeX: \displaystyle f'(x) = 5^{5^{v}} 5^{v} 5^{x} \ln{\left(5 \right)}^{3} = 5^{5^{5^{x}}} 5^{5^{x}} 5^{x} \ln{\left(5 \right)}^{3}$ .