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

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