The authors analyze different definitions of fractional derivatives and their applications to fractional differential equations and system theory. First, a set of criteria for an operator to be a FD is given. Next, the \(\alpha\)-order derivative is defined as the convolution operator \[ D^{\alpha}_{\theta}f(t)=\int _R f(t-\tau )\psi^{\alpha}_{\theta} (\tau )\, d\tau, \] where \(\psi^{\alpha}_{\theta} (t)\) with \(t\in R\) is the kernel of the derivative and \(\theta \in R\) is an asymmetry parameter that controls the characteristics of the derivative, namely the causality. The kernel has the Fourier transform \(\vert \omega \vert ^{\alpha} e^{i\theta \frac{\pi}{2}sgn (\omega)}\) such that the corresponding Bode diagram is a straight line. The explicit form of the kernel is given. The convolutional operator has the properties that it is linear, has sinusoids as eigenfunctions, has the properties of additivity and commutativity. The inverse operator is given. In the causal case when \(\theta = \alpha\), one gets the Liouville and Liouville-Caputo derivatives. Specific examples are considered. First, the CF-operator (defined by Caputo-Fabrizio) is given and analyzed. Because the Bode-operator then has two broken lines the conclusion is that this is neither a fractional nor a derivative operator. Next, the exponential in the CF-operator is replaced by the Mittag-Leffler function. This gives the AB-operator. This operator is fractional but not a derivative. The importance of the additivity and commutativity properties is dealt with in more detail. In particular with their importance in system theory. A state-space equation is considered.