A general method of defining what it means to take the one half derivative and the one half integral of a function is discussed. A consistent mathematics that works for any function of exponential order is developed using Laplace Transform theory that works for any fractional order of differentiation and integration.

Some Laplace Transform pairs are introduced that are instrumental in solving for the fractional derivative of polynomials. The gamma function and some useful identities are also presented with examples.

Part 2 takes the discussion further with the solution to a 1/2 order differential equation:

https://www.youtube.com/watch?v=SW67hqd4P_w

Check the time index in the pinned comment below for a specific topic you may be interested in.

For application of fractional derivatives refer to:

https://en.wikipedia.org/wiki/Fractional_calculus#Applications