Introduction
Derivative of “exp” means a derivative of $e^x$, where the number $e$ is an Euler’s constant, roughly equal to $2.71828$. As an equation,
\[\begin{align*} \exp(x) = e^x, \text{where e} \approx 2.71828. \end{align*}\]
Derivative of $e^x$
The derivative of the function is equal to the slope of the function at a given point. The derivative of $e^x$ is unique because it is equal to the function itself. This is the same as saying that
\[\begin{align*} \boxed{ \dfrac{d}{dx}(e^x) = e^x.} \end{align*}\]
For example, if the position of a car is given by $e^t$, then its velocity is also $e^t$.
However, if the exponent is not just $x$ but some expression in terms of $x$ then the derivative of such an exponential function might not equal the function itself. For example, if we’re taking the derivative of $f(x) = 2e^{2x}$, then:
\[\begin{align*} \dfrac{d}{dx}\left(f(x)\right) = \dfrac{d}{dx}(e^{2x}) = 2 \cdot e^{2x} \end{align*}\]
