Show that the function f x y xy 8y2 is differentiable by

Last updated: 6/29/2023

Show that the function f x y xy 8y2 is differentiable by finding values of and 2 that satisfy the following definition Definition If z f x y then f is differentiable at a b if Az can be expressed in the form Az f a b Ax fy a b Ay E AX E2Ay where and 2 0 as Ax Ay 0 0 To find an expression for Az we choose an arbitrary point a b another point arbitrarily close to the first a Ax b Ay and compute the difference between their respective f values Az f a Ax b Ay f a b a Ax b Ay 8 b Ay ab aAy b x AxAy 8b2 16bAy 8 Ay 2 ab Ay bAx AxAy but fx a b and f a b Az f a b Ax f a b Ay AxAy 8AyAy Let Ay and 2 8Ay therefore f is differentiable by the given definition AyAy SO