Discover all the functions' second-order partial derivatives.&nbsp;f(x,y)=sin⁡xy

Step 1 the given function: f(x,y)=sin⁡xy we have to find the second order partial derivatives of the given function: i.e. fxx,fyy,fxy,fyx. Step 2 here, f(x,y)=sin⁡(xy) firstly , we find fx,fy fx=∂f∂x=ycos⁡xy and fy=∂f∂y=xcos⁡xy now we have to find fxx,fyy fxx=∂fx∂x=−y2sin⁡xy and fyy=∂fy∂fy=−x2sin⁡xy now we have to find fxy,fyx fxy=∂fx∂y=−yxsin⁡xy and fyx=∂fy∂x=−xysin⁡xy this is the required answer.

Question: "Discover all the functions' second-order partial derivatives.f(x,y)=sin⁡xy"

coexpennan

Answered question

2021-06-06

Discover all the functions' second-order partial derivatives. $f (x, y) = \sin x y$

Answer & Explanation

Neelam Wainwright

Skilled2021-06-07Added 102 answers

Step 1
the given function:
$f (x, y) = \sin x y$
we have to find the second order partial derivatives of the given function:
i.e. $f_{x x}, f_{y y}, f_{x y}, f_{y x}$ .
Step 2
here, $f (x, y) = \sin (x y)$
firstly , we find $f_{x}, f_{y}$
$f_{x} = \frac{\partial f}{\partial x} = y \cos x y and f_{y} = \frac{\partial f}{\partial y} = x \cos x y$
now we have to find $f_{x x}, f_{y y}$
$f_{x x} = \frac{\partial f_{x}}{\partial x} = - y^{2} \sin x y and f_{y y} = \frac{\partial f_{y}}{\partial f_{y}} = - x^{2} \sin x y$
now we have to find $f_{x y}, f_{y x}$
$f_{x y} = \frac{\partial f_{x}}{\partial y} = - y x \sin x y and f_{y x} = \frac{\partial f_{y}}{\partial x} = - x y \sin x y$
this is the required answer.

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