Let f be a continuous function on R, and define
A(x)=∫g(x)h(x)f(t)dt,
where g and h are differentiable functions on an interval I.
Throughout the question, assume that h(x)=g(x) for all x∈I.
(a) Starting from the definition of the derivative, write the limit expression for A′(x) and describe—in words and symbols—the first algebraic step needed to transform the single integral in the numerator into a sum or difference of two integrals with a common lower limit.
(b) Use your work in part (a) to obtain an explicit formula for A′(x) in terms of f, g, h, and their derivatives. Provide a clear justification for each term that appears in your formula.
(c) Suppose h(x)<g(x) for every x in a sub-interval J⊂I. Explain how your formula in part (b) predicts the sign of A′(x) on J when f is known to be strictly positive on the range of integration.