$e$ is Euler's number, about 2.71828, read aloud simply e; it is the one base for which the exponential curve's slope equals its height at every point: $\dfrac{d}{dx}\,e^x = e^x$, read aloud: the derivative of e to the x is e to the x. Here $x$ is the exponent, the variable, and $e$ is a fixed number; the power rule $x^n \to nx^{n-1}$ is for a variable BASE with a fixed exponent, and forcing it onto $e^x$ to get $x e^{x-1}$ is the trap in the lesson's name. With a multiple of $x$ upstairs, the chain rule runs with $u = bx$: $\dfrac{d}{dx}\,e^{bx} = b\,e^{bx}$, b e to the b x, the function itself times the inner slope $b$. So $5e^{0.2x}$ has derivative $1 \cdot e^{0.2x}$, and at $x = 10$ that is $e^2 = 7.38906$.

$\ln c$, read aloud natural log of c, is the power $e$ must be raised to in order to give $c$: $e^{\ln c} = c$ by definition, and $e^{b \ln c} = c^b$ by the log laws. That is what makes exponentials clean at the points $x = \ln c$: for $f(x) = 3e^{2x}$, $f'(x) = 6e^{2x}$, and $f'(\ln 2) = 6 \cdot e^{2 \ln 2} = 6 \cdot 2^2 = 24$, no calculator. At $x = 0$, $e^0 = 1$, so the slope of $k e^{bx}$ there is just $kb$. Anything to the power 0 is 1, not 0 and not $e$.