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Integrals Depending on a Parameter ft is also useful to know how to differentiate with respect to some parameter that affects either the limits of an integral or the integrand. The economic applications of these techniques are primarily in the field of dynamic analysis (see chapter 25), and it is traditional to use the variable t. which represents time, as the variable of integration. We will usex as the parameter which may affect either the integrand or the limits of integration.

First, consider the case in which only the upper limit of an integral, U. depends on the parameter, x, which we will write as Fix) = j'J " f(t)dt. By using the result of theorem 16.2 and the chain rule for differentiat on. we obtain the resuh that

du ox tlx

Recall the intuition from theorem 16.2. That part of the above result giving dF/'dU = fiV) implies lhat a marginal increase in the value of the upper limit increases the area under the curve described by fit) at a rate that equals the height of the curve at that point, which is simply /(£/). Similarly, if the lower limit depends on the parameter x, writing this as L(x), then a marginal increase in L of one unit decreases the area under the curve between the two limits of integration by the amount /(L i. Therefore it follows that if Fix) = //',, ] fit) dt. then where the minus sign indicates that the area is reduced by an increase in L (or is increased by a decrease in /.).

Now suppose diat it is the integrand that depends on the parameter x, which we write as F(x) = f f(t,x\dt. The following result is often referred to as Leibniz's rule:

We do not provide a formal proof for this result, but the intuition is clear. The expression on the right-hand side of the equality essentially measures the change in the area under the curvc /([, .v) created by changing the value of the parameter x by some small marginal unit. The rate of change. F'{x). is found by integrating over the rate of change in f(t, x) with respect to a change in x between the limits of integration.

We can summarize these three results for the case where F(x) depends on the parameter* as a result of both limits of integration depending on x and the integrand depending on x: namely Fix) — //,' ,' f(l \)dt, This gives us the result

<$U ic tif(t.x) , F'(x) = -f(L.X)- — + f(U,x)- — + / —-dt ox ax Jl ox

A further generalization of this result is that if the integrand depends on a parameter z. which in turn depends on the parameter v, then we can write F( v) -f'nW' '(x))dt, and applying the chain rule of differentiation, we obtain ol w fv dftr, z) dz ,

F (x) -- f(l.z) ■ — + f(U, z) • — I- I — di ox rfjf Ji Bz ox

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