For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step. f(x,y)=x2y−xy2,where x=standy=st

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For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step. f(x,y)=x2y−xy2,where x=standy=st

Asma Vang · Accepted Answer

Step 1 Given function is f(x,y)=x2y−xy2 where x=st,y=s. The all relevant second derivatives are fxx,fyy,fxy,fyx. Step 2 Find partial derivative with respect to x for the function f(x,y)=x2y−xy2. ⇒fx=2xy−y2 Find partial derivative with respect to y for the function f(x,y)=x2y−xy2. ⇒fy=x2−2xy Differentiate function fx=2xy−y2 withe respect to x to get f×. ⇒fxx=2y Substitute y=st∈fxx=2y. ⇒fxx=2st Differentiate function fy=x2−2xy with respect to y to get fyy. ⇒fyy=−2x Substitute x=st in fyy=−2x. ⇒fyy=−2st Differentiate function fy=x2−2xy with respect to x to get fxy. ⇒fxy=2x−2y Substitute x=st,y=s∈fxy=2x−2y. ⇒fxy=2st−2st Differentiate function fx=2xy−y2 with respect to y to get fyx. ⇒fyx=2x−2y Substitute x=st,y=s∈fyx=2x−2y ⇒fyx=2st−2st. Step 3 Therefore, all relevant second derivatives of function

Jeffrey Jordon · Accepted Answer

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For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step. f(x,y)=x^2y-xy^2,where x = st and y = s/t

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