Implication of exponential growth: how is it deduced? Let $L$ be a differentiable function defined on $\mathbb{R}\times\Omega$ with $\Omega\subseteq\mathbb{R}^n$. I will say it has _exponential growth_ if for all $O\subset\subset\Omega$ open there exists a constant $C(O)$ such that: $$|\nabla_\xi L(x,\xi)|\leq C(O)L(x,\xi),$$ for all $\xi\in O$. A photocopy I was given from a book I can't identify uses this condition to deduce the following: $$\frac{d}{dt}L(x,a+tb)=\langle\nabla_\xi L(x,a+tb),b\rangle\leq c|b|L(x,a),$$ and I understand that, and then: $$L(x,a+b)\leq L(x,a)e^{c|b|}.$$ How is this last inequality deduced from the previous one?

Integrate the inequality from $t = 0$ to $t=1$ gives

$$L(x, a+b) - L(x, a) \le c|b| L(x,a) \Rightarrow L(x, a+b) \le (c|b| +1) L(x,a).$$

Now the result follows the inequality $e^x \ge 1+x$ for $x\ge 0$.