Find the derivative of $LaTeX: \displaystyle f(x) = \sin{\left(e^{4 x} \right)}$ .

Decomposing the function gives $LaTeX: \displaystyle f(u) = \sin{\left(u \right)}$ , $LaTeX: \displaystyle u = v^{4}$ , and $LaTeX: \displaystyle v = e^{x}.$ Using the chain rule $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$ . $LaTeX: \displaystyle f'(x) = (\cos{\left(u \right)})(4 v^{3})(e^{x}) = 4 v^{3} e^{x} \cos{\left(u \right)}$ . Substituting back in $LaTeX: \displaystyle u$ and $LaTeX: \displaystyle v$ gives $LaTeX: \displaystyle f'(x) = 4 v^{3} e^{x} \cos{\left(v^{4} \right)} = 4 e^{4 x} \cos{\left(e^{4 x} \right)}$ .