Existence of a differentiable function satisfying a condition Show that there exists a real number $\epsilon>0$ differentiable function $f:(-\epsilon, \epsilon)\rightarrow \mathbb{R}$ such that
$$e^{x^2+f(x)}=1-\sin(x+f(x)).$$
I have no idea about how to proceed. Any ideas?

Hint : apply the Implicit function theorem to the function
$$F(x,y)=e^{x^2+y}-1+\sin(x+y).
$$
