Find the derivative of $LaTeX: \displaystyle f(x) = \sin{\left(\left(x^{3}\right)^{\frac{7}{2}} \right)}$ .

Decomposing the function gives $LaTeX: \displaystyle f(u) = \sin{\left(u \right)}$ , $LaTeX: \displaystyle u = v^{\frac{7}{2}}$ , and $LaTeX: \displaystyle v = x^{3}.$ Using the chain rule $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$ . $LaTeX: \displaystyle f'(x) = (\cos{\left(u \right)})(\frac{7 v^{\frac{5}{2}}}{2})(3 x^{2}) = \frac{21 v^{\frac{5}{2}} x^{2} \cos{\left(u \right)}}{2}$ . Substituting back in $LaTeX: \displaystyle u$ and $LaTeX: \displaystyle v$ gives $LaTeX: \displaystyle f'(x) = \frac{21 v^{\frac{5}{2}} x^{2} \cos{\left(v^{\frac{7}{2}} \right)}}{2} = \frac{21 x^{2} \left(x^{3}\right)^{\frac{5}{2}} \cos{\left(\left(x^{3}\right)^{\frac{7}{2}} \right)}}{2}$ .