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

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