Let φ(a) = ∫_a^(a+100π) |sin x| dx. What is φ'(a) equal to?

Since φ(a) = 200 is a constant (independent of a), its derivative is φ'(a) = 0. Alternatively, by Leibniz rule, φ'(a) = |sin(a+100π)| - |sin a| = |sin a| - |sin a| = 0, since sin has period 2π and 100π is a multiple of 2π.