Let : [a, b] [A, B] be such that 0 is positive and continuous

Let α : [a, b] → [A, B] be such that α 0 is positive and continuous on [a, b]. Let β : [A, B] → [a, b] be the inverse function of α and let f : [a, b] → R be continuous. 

Define F(y) = Z β(y) a f(t) dα(t) (A ≤ y ≤ B). 

Prove that F 0 (y) = (f ◦ β)(y) for all y ∈ [A, B].  Hint: β 0 (y) = 1 α0 β(y)
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