# Exponential Fourier transform and derivative

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?

$$\dfrac{d}{dt}e^{-\alpha t}u(t) = e^{-\alpha t}\delta(t) -\alpha e^{-\alpha t}u(t)$$