Understanding Leibniz's Rule: Use, Why & Examples

[glebovg]

Can anyone explain Leibniz's rule?

When would one use it and why?

Examples?

 

[Stephen Tashi]

Which Leibniz rule? How to differentiate a definite integral when the variable appears as limit of integration? How to differentiate the nth power of a product of functions?

 

[glebovg]

Leibniz integral rule.

 

[Stephen Tashi]

You might find a forum member with a pent-up enthusiasm for explaining the Leibniz integral rule. Failing at that, I suggest you ask a more specific question. After all, there are all sorts of treatments of the Leibniz integral rule on the web. What answers one persons "why?" may not satisify another. What do you want to see? a physics problem that uses it? the usual calculus of variations stuff? how to work tricky integrals with it?

 

[jackmell]

When would one use it and why?

Examples?

Oh man like all the time in math just doing regular stuff. Just a few days ago I ran into this real-life problem:

If:

$$k\int_c^d \frac{e^{b\sqrt{s^2-a^2}}}{\sqrt{s^2-s^2}}e^{sx}ds=I_0(a \sqrt{x^2-b^2})$$

then find:

$$k\int_c^d e^{b\sqrt{s^2-a^2}}e^{sx}ds$$

Now, look up Leibniz rule, then differentiate the first expression with respect to b, and find the solution to the second integral.

 

[stevenb]

Can anyone explain Leibniz's rule?

When would one use it and why?

Examples?

The other day, I was reading a paper that claimed that

$$x(t)=x(0) \exp (at)+b\int_0^t \exp(a(t-\tau))\; u(\tau) d\tau$$

was a solution to

$${{dx(t)}\over{dt}}=a\; x(t)+b\; u(t)$$

I wanted to quickly verify this and used the integral rule to check it. I also double checked it with a Laplace transform method, just for practice, but that's not relevant here.

The integral here is a convolution integral, and differentiating a convolution integral is one example where the rule comes in handy. Note that the derivative with respect to t can not just be moved into the integral without additional modifications because the upper limit depends on t. The Leibniz rule, let's you write the answer correctly and quickly. It does this by adding terms that depend on the derivatives of the limits.