Show that the function f(x, y) = xy – 6y2 is differentiable by finding values of ɛ, and ɛ, 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 + f,(a, b) Ay + 8,Ax + 8,Ay, where s, and 8, → 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 + Дх, b + Ду) - f(a, b) %3D (а + Дх)(b + Ду) - 6(Ь + Дy)? - = ab + aly + bAx + AXAY – 6b2 - 12bay - 6(Ay)2 - ab + Ay + bAx + AxAy - AyA but f,(a, b) = and f(a, b) = , So Az = f,(a, b)Ax + f,(a, b)Ay + AxAy – 6AyAy. Let ɛ, = Ay and ɛ, = -6Ay, therefore f is differentiable by the given definition.

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