  We present exact analytical results for the Caputo fractional derivative of a
wide class of elementary functions, including trigonometric and inverse
trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian,
and Lorentzian functions. These results are especially important for
multi-scale physical systems, such as porous materials, disordered media, and
turbulent fluids, in which transport is described by fractional partial
differential equations. The exact results for the Caputo fractional derivative
are obtained from a single generalized Euler's integral transform of the
generalized hyper-geometric function with a power-law argument. We present a
proof of the generalized Euler's integral transform and directly apply it to
the exact evaluation of the Caputo fractional derivative of a broad spectrum of
functions, provided that these functions can be expressed in terms of a
generalized hyper-geometric function with a power-law argument. We determine
that the Caputo fractional derivative of elementary functions is given by the
generalized hyper-geometric function. Moreover, we show that in the most
general case the final result cannot be reduced to elementary functions, in
contrast to both the Liouville-Caputo and Fourier fractional derivatives.
However, we establish that in the infinite limit of the argument of elementary
functions, all three definitions of a fractional derivative - the Caputo,
Liouville-Caputo, and Fourier- converge to the same result given by the
elementary functions. Finally, we prove the equivalence between
Liouville-Caputo and Fourier fractional derivatives.
