Die Funktion f(x, y) sei auf [a, b] ×[c, d] definiert, für jedes feste x ∈[a, b] sei sie R-integ

Die Funktion f(x, y) sei auf [a, b] ×[c, d] definiert, für...

Key Concepts

Riemann Integration

Riemann integration is a method for assigning a number to the area under a curve defined by a function on an interval. It formalizes the concept of integration by partitioning the domain into subintervals, summing the contributions over these partitions, and taking the limit as the partition becomes arbitrarily fine. This concept is essential when discussing the integrability of functions with respect to one variable while another parameter is held fixed.

Partial Differentiation

Partial differentiation involves computing the derivative of a multivariable function with respect to one of its variables while holding the others constant. This concept is key when analyzing the sensitivity or rate of change of a function in one specific direction, which is important in many applications including the analysis of functions defined over product spaces.

Differentiation Under the Integral Sign

Differentiation under the integral sign, also known as the Leibniz rule, is a technique that allows one to differentiate an integral whose limits or integrand depends on a parameter. Under certain conditions such as the continuity or boundedness of the integrand's derivative, it is permissible to interchange the order of differentiation and integration. This technique is particularly useful in evaluating parameter-dependent integrals and solving complex integrals by differentiating them with respect to a parameter.

Boundedness Conditions

Boundedness conditions require that a function or its derivative does not exceed a certain finite value over its domain. In the context of differentiating under the integral sign, ensuring that the partial derivative of the integrand is bounded is critical. It guarantees that the exchange of differentiation and integration is valid and that the resulting integral of the derivative is well-defined and manageable.

Transcript

0:00 Hi there.

00:01 In this problem, we're asked to find the partial derivatives of this function here with respect to x and y.

00:07 So let's begin with x.

00:11 Now, the first thing to notice is that we have a product here.

00:15 We have one function of x and y, in this case it's just of x, times another function of x and y.

00:24 So we'll use the product just as we always have.

00:28 So we know what the product rule says.

00:30 It depends what order you tend to do it in.

00:32 It doesn't matter.

00:32 Let's not.

00:33 Just do it in this order where we have the first function as it is, no derivative, times the derivative of the second function.

00:45 Now, be careful.

00:46 When we say derivative now, we mean the partial derivative with respect to x.

00:51 So the partial with respect to x of this second function.

00:55 Okay.

00:58 Now, plus, now again, according to the product rule, we need now the first derivative.

01:04 I'm sorry, we need the derivative of the first function here.

01:08 And again, when we say derivative, we mean the partial with respect to x of e to the minus x, and then times just the second function as it is no derivative.

01:20 So it's just the product tool we've always known and used.

01:23 The only difference is when we take derivatives now, it matters which variable we're taking the partial derivative with respect to.

01:29 In this case, it's x.

01:31 Okay.

01:33 Let's see.

01:34 Let's actually take this derivative now.

01:36 The derivative of sine of x plus y with respect to x.

01:41 Notice we'll need a chain rule here since we have a function inside of a function.

01:46 So keep everything blue that's part of this derivative.

01:50 The derivative of sine is cosine.

01:54 We leave the argument, x plus y, just as we always do.

01:58 We keep that the same.

01:59 But now by the chain rule, we need to multiply by the derivative of this inner function, the x plus y.

02:06 And again, when we say derivative, what we mean is the partial with respect to x of x plus y.

02:17 Okay.

02:19 Now over here, this is a little easier.

02:21 The partial with respect to x of e to the minus x, there's not even a y in here.

02:26 So this will be just like we've always been used to for many, many months now.

02:30 The derivative here, we get e to the x or e to the minus x.

02:34 And then by the chain rule, we need to multiply by the derivative of minus x, which is minus 1.

02:38 So we put a minus in front.

02:42 And we'll keep sine of x plus y there.

02:44 Okay.

02:46 So the only thing we really have to do still, i'm just going to keep everything we have.

02:53 We still have to take this derivative.

02:55 This finally is a little faster.

02:57 The partial derivative of x plus y with respect to x...

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