Derivative of a Scaled Exponential Function

Let $c$ be a real number and define $E_c:\mathbb{R}\to\mathbb{R}$ by $E_c(t)=\exp(ct)$, with the exponential function. The set $\mathbb{R}$ is an interval and every real number is an interior point of it. Then $E_c$ is differentiable at every $t\in\mathbb{R}$ with

and $E_c$ is continuous at every $t\in\mathbb{R}$.