Find the derivative of $LaTeX: \displaystyle f(x) = 3^{3^{3^{x}}}$

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