Find the derivative of $LaTeX: \displaystyle f(x) = 3^{2^{7^{x}}}$

Taking the derivative with the chain rule gives $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$ . $LaTeX: \displaystyle f'(x) = (3^{u} \ln{\left(3 \right)})(2^{v} \ln{\left(2 \right)})(7^{x} \ln{\left(7 \right)})$ . Substituting back in $LaTeX: \displaystyle u$ and $LaTeX: \displaystyle v$ gives $LaTeX: \displaystyle f'(x) = 2^{v} 3^{2^{v}} 7^{x} \ln{\left(2 \right)} \ln{\left(3 \right)} \ln{\left(7 \right)} = 2^{7^{x}} 3^{2^{7^{x}}} 7^{x} \ln{\left(2 \right)} \ln{\left(3 \right)} \ln{\left(7 \right)}$ .