Calculate all four second-order partial derivatives and check that fx y=fy x . Assume the variab

Calculate all four second-order partial derivatives and...

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions, where one function is applied to the result of another function. In the context of the function f(x, y) = sin(x² + y²), it helps in differentiating sin(u) with respect to x or y by first differentiating sin(u) with respect to u and then multiplying by the derivative of u = x² + y² with respect to the corresponding variable.

Partial Derivatives

Partial derivatives measure the rate at which a function changes with respect to one variable while keeping the other variables constant. For the given function, first-order partial derivatives with respect to x and y are computed by treating one variable as constant and using differentiation rules, including the chain rule.

Second-Order Partial Derivatives

Second-order partial derivatives involve differentiating a function twice. They can be either pure derivatives, where the same variable is differentiated twice, or mixed derivatives, where the differentiation is done with respect to different variables. These derivatives provide insight into the curvature and concavity of the function in various directions.

Mixed Partial Derivatives

Mixed partial derivatives are the results of differentiating a function first with respect to one variable and then with respect to another. In this function, f_xy represents differentiating first with respect to x and then y, while f_yx involves differentiating first with respect to y and then x. Mixed partial derivatives are particularly useful in studying the behavior of multivariable functions and verifying conditions of smoothness.

Clairaut's Theorem

Clairaut's theorem, also known as Schwarz's theorem, states that if the mixed partial derivatives of a function are continuous in a region, then the order of differentiation does not matter, and f_xy equals f_yx. Verifying this equality for a function like f(x, y) = sin(x² + y²) serves as an application of this theorem, confirming the function's smoothness and the consistency of its second-order behavior.

Transcript

00:01 In this question, we will be talking about second order partial derivatives.

00:07 The question asks us to find these second order partial derivatives for this particular bivariate function.

00:16 So to find the second partial derivatives, we need the first partial derivatives.

00:22 And so let's differentiate with respect to x.

00:28 To do this, we have to keep in mind that x is, there's a function applied to x, the squared function and that squared function is also within another function, sign function.

00:42 And so we must use chain rule.

00:45 That is, we differentiate the sine function and we multiply this by the derivative of the squared function.

01:01 And so there is our derivative with respect to x.

01:04 Now to find the second derivatives that come from this, we differentiate this derivative with respect to x itself with respect to y and with respect to x so first let's do the derivative of this derivative respect to y so if we differentiate with respect to y this is treated as a constant and we just differentiate this and stick this constant onto the derivative now the derivative of the coast function is negative sign and since we have the squared function applied to y inside of this function we must multiply this by the derivative of this squared function.

01:57 But we also need to stick this on, like i said.

02:03 So, and so there we have this second derivative.

02:10 Now, if we differentiate this derivative that's of the first order, now all with respect to x, we differentiate with respect to x once more.

02:20 Then we have to use the product rule, because we have two functions of x.

02:27 So first we take the, uh, uh, differentiate the first function and multiply the second and then we differentiate this function, the second one, and multiply that derivative by the 2x...

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