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

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