Find the derivative of $LaTeX: \displaystyle f(x) = \cos{\left(\sin{\left(x^{5} \right)} \right)}$ .

Decomposing the function gives $LaTeX: \displaystyle f(u) = \cos{\left(u \right)}$ , $LaTeX: \displaystyle u = \sin{\left(v \right)}$ , and $LaTeX: \displaystyle v = x^{5}.$ Using the chain rule $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$ . $LaTeX: \displaystyle f'(x) = (- \sin{\left(u \right)})(\cos{\left(v \right)})(5 x^{4}) = - 5 x^{4} \sin{\left(u \right)} \cos{\left(v \right)}$ . Substituting back in $LaTeX: \displaystyle u$ and $LaTeX: \displaystyle v$ gives $LaTeX: \displaystyle f'(x) = - 5 x^{4} \sin{\left(\sin{\left(v \right)} \right)} \cos{\left(v \right)} = - 5 x^{4} \sin{\left(\sin{\left(x^{5} \right)} \right)} \cos{\left(x^{5} \right)}$ .