Deriving from Doppler's formula

I am studying Doppler sonography. I do understand doppler's formula: $$f=f_{0}\frac{v}{v-u},$$ where $$f_{0}$$ is emitted frequency, f is received frequency, v is speed of the ultrasound puls and u is speed of blood from which is the ultrasound puls bouncing back. But I am struggling with how to get from above formula to this: $$\Delta f = f-f_{0}=2f_{0}u.$$ Can anyone one give me hint? Thank you.

Image I am working with

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• That does not look right. $f-f_0 = f_{0}\frac{v}{v-u} - f_0 = f_{0}\frac{u}{v-u}$ is not equal to $2 f_0 u$. Jun 23 at 7:46
• Then I misinterpreted the image I added to my question probably. Jun 23 at 8:11
• It looks like the image is trying to sat $\Delta F=2F_Tv/c$.
– J.G.
Jun 23 at 8:15
• Ok, this makes more sense.. but still is this formula directly derived from $$f=f_{0}\frac{v}{v-u}?$$ Jun 23 at 8:26