Calculus (3rd Edition)

$$\frac{e}{2}.$$
By differentiation, we have $$f'(g(x))g'(x)=2xe^{x^2}$$ hence we have $$f'(g(1))g'(1)=f'(2) 4=2e \Longrightarrow f'(2)=\frac{e}{2}.$$