Leibniz’s Rule In applications, we sometimes encounter functions like defined by integrals that have variable upper limits of integration and variable lower limits of integration at the same time. The first integral can be evaluated directly, but the second cannot. We may find the derivative of either integral, however, by a formula called Leibniz’s Rule.
If ƒ is continuous on [a, b] and if u(x) and (x) are differentiable functions of x whose values lie in [a, b], then
Figure 5.33 gives a geometric interpretation of Leibniz’s Rule. It shows a carpet of variable width ƒ(t) that is being rolled up at the left at the same time x as it is being unrolled at the right. (In this interpretation, time is x, not t.) At time x, the floor is covered from u(x) to (x). The rate at which the carpet is being rolled up need not be the same as the rate at which the carpet is being laid down. At any given time x, the area covered by carpet is
FIGURE 5.33 Rolling and unrolling a carpet: a geometric interpretation of Leibniz’s Rule:
At what rate is the covered area changing? At the instant x, A(x) is increasing by the width ƒ( (x)) of the unrolling carpet times the rate at which the carpet is being unrolled. That is, A(x) is being increased at the rate
At the same time, A is being decreased at the rate
the width at the end that is being rolled up times the rate . The net rate of change in A is
which is precisely Leibniz’s Rule.
To prove the rule, let F be an antiderivative of ƒ on [a, b]. Then.
Differentiating both sides of this equation with respect to x gives the equation we want:
Use Leibniz’s Rule to find the derivatives of the functions.
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