Question

Verify the conclusion of Clairut's theorem for the following function. That is, show that f xy=fyx for f(x,y)=sin(x 2y)(3) Find f xyx and f yxy of f(x,y)=sin(2x+5y) (Note that you can use Clairut's theorem at some point, but you don't have to use it. It will just save you some time if you use it correctly).

Answer 1

To verify Clairaut's theorem for the given function f(x, y) = sin(x^2y)(3), we need to find fxyx and fyxy. Applying the chain rule, we find the second-order mixed partial derivatives.

Step-by-step explanation:

To verify the conclusion of Clairaut's theorem, we need to find the second-order mixed partial derivatives of the given function f(x, y) = sin(x^2y)(3).

To find fxyx, we differentiate the function twice with respect to x and once with respect to y. Applying the chain rule, we get fxyx = (2xy cos(x^2y)+3x^2 sin(x^2y))(3).

To find fyxy, we differentiate the function twice with respect to y and once with respect to x. Again, applying the chain rule, we get fyxy = (x^4 cos(x^2y))(3).

Therefore, fxy = fyx for the given function.