Expression for $\exp\left(x\right) -1$

Question

Found an unexpected expression: $$e^x -1=xe^{\theta.x}$$ where $\theta \in \left(0,1\right)$. However, I cannot prove it (the $\theta$ part). Any ideas? Thank you.

Answer 1

Thank you greatly to everybody for your help. Mean Value Theorem works:

Fix x and consider the function $f(y)=\exp \left(xy\right)$ on the interval $y \in \left(0, 1\right)$. The function $f\left(y\right)$ is differentiable on this interval with respect to $y$. Then, by the MVT, there exists $\theta \in \left(0, 1\right)$ such that $$x \exp \left(x \theta\right)=\exp \left(x\right)-1.$$ Done.