Find the derivative of $LaTeX: \displaystyle f(x) = 4^{4^{4^{x}}}$

Taking the derivative with the chain rule gives $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$ . $LaTeX: \displaystyle f'(x) = (4^{u} \ln{\left(4 \right)})(4^{v} \ln{\left(4 \right)})(4^{x} \ln{\left(4 \right)})$ . Substituting back in $LaTeX: \displaystyle u$ and $LaTeX: \displaystyle v$ gives $LaTeX: \displaystyle f'(x) = 4^{4^{v}} 4^{v} 4^{x} \ln{\left(4 \right)}^{3} = 4^{4^{4^{x}}} 4^{4^{x}} 4^{x} \ln{\left(4 \right)}^{3}$ .