Find the derivative of $LaTeX: \displaystyle f(x) = 8^{\cos{\left(\sin{\left(x \right)} \right)}}$ .

Decomposing the function gives $LaTeX: \displaystyle f(u) = 8^{u}$ , $LaTeX: \displaystyle u = \cos{\left(v \right)}$ , and $LaTeX: \displaystyle v = \sin{\left(x \right)}.$ Using the chain rule $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$ . $LaTeX: \displaystyle f'(x) = (8^{u} \ln{\left(8 \right)})(- \sin{\left(v \right)})(\cos{\left(x \right)}) = - 8^{u} \ln{\left(8 \right)} \sin{\left(v \right)} \cos{\left(x \right)}$ . Substituting back in $LaTeX: \displaystyle u$ and $LaTeX: \displaystyle v$ gives $LaTeX: \displaystyle f'(x) = - 8^{\cos{\left(v \right)}} \ln{\left(8 \right)} \sin{\left(v \right)} \cos{\left(x \right)} = - 8^{\cos{\left(\sin{\left(x \right)} \right)}} \ln{\left(8 \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}$ .