Derivatives of Exponential Derivatives

Use the definition of $e$ as the unique positive number for which $\lim_{h\rightarrow 0}\frac{e^{h} - 1}{h} = 1$ and the definition of the derivative to show that derivative of the exponential function, $f(x) = e^x$ is equal to $e^x$

Find the derivative of $f(x) = x^{3}e^{-2x}$

Determine the slope of the tangent line to the function $f(x) = 2e^{-3x}$at $(0,2)$

For the following problem, find the derivative of $f(x) = 5^{x^{3} - 4}$