  The quantum algorithms for Monte Carlo integration (QMCI), which are based on
quantum amplitude estimation (QAE), speed up expected value calculation
compared with classical counterparts, and have been widely investigated along
with their applications to industrial problems such as financial derivative
pricing. In this paper, we consider an expected value of a function of a
stochastic variable and a real-valued parameter, and how to calculate
derivatives of the expectation with respect to the parameter. This problem is
related to calculating sensitivities of financial derivatives, and so of
industrial importance. Based on QMCI and the general-order central difference
formula for numerical differentiation, we propose two quantum methods for this
problem, and evaluate their complexities. The first one, which we call the
naive iteration method, simply calculates the formula by iterative computations
and additions of the terms in it, and then estimates its expected value by QAE.
The second one, which we name the sum-in-QAE method, performs the summation of
the terms at the same time as the sum over the possible values of the
stochastic variable in a single QAE. We see that, depending on the smoothness
of the function and the number of qubits available, either of two methods is
better than the other. In particular, when the function is nonsmooth or we want
to save the qubit number, the sum-in-QAE method can be advantageous.
