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

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