The Fourier transform of right handed exponential function $e^{-\alpha t}$ is given by
$$\mathcal{F}\left[ e^{-\alpha t}\right] = \frac{1}{\alpha + j \omega}.$$
Its derivative is
$$\frac{d}{dt} e^{-\alpha t} = -\alpha e^{-\alpha t}.$$
So the Fourier transform of derivative is
$$\mathcal{F}\left[ \frac{d}{dt}e^{-\alpha t}\right] = -\frac{\alpha}{\alpha + j \omega}.$$
But if I use the Fourier transform derivative formula
$$j \omega F(\omega) = \frac{j \omega}{\alpha + j \omega}.$$
These results are different. Where is my mistake?
Your mistake is that the derviative is
$$\dfrac{d}{dt}e^{-\alpha t}u(t) = e^{-\alpha t}\delta(t) -\alpha e^{-\alpha t}u(t)$$
From there, the rest follows and agrees with the derivative theorem of the Fourier transform.