Which of the following is a correct relation for a transistor

$I_E = I_B + I_C$

Differentiating, we get

$ΔI_E = ΔI_B + ΔI_C$

$⇒\frac{ΔI_E}{ΔI_C}=\frac{ΔI_B}{ΔI_C}+1$

$⇒\frac{1}{α}=\frac{1}{β}+1=\frac{1+β}{β}$

$⇒α=\frac{β}{1+β}$