Q.13 - Calculus

Question

What does it mean for a function of two variables, $f(x,y),$ to be differentiable at a point$(a,b)$ ?

Step-by-Step Solution

Answer

Differentiability at the point of function for two variables is$\Delta x$ and $∆y,$ and both goes to zero as $(∆x,∆y) \to (0,0).$

Step 1 : Introduction

Differentiability for function of two variables

$f(x, y)$ be a function of two variables defined on an open set containing the point $(a, b).$and let $\nabla f(x,y)=f(a+\Delta x,b+\Delta y)-f(a,b)$ The function $f$is said to be differentiable at          

  $(a, b)$ if the partial derivative$f_x(a,b)$ and $f_y(a,b)$ both exist and

$\nabla f(x,y)=f_x(a,b)\Delta x+f_y(a,b)\Delta y+{ϵ}_1\Delta x+{ϵ}_2\Delta y$

 

Step 2 : Solution

Where ${\epsilon }_1$ and ${\epsilon }_2$ are function of $\Delta x$ and $\Delta y$, and both goes to zero as $(\Delta x,\Delta y)\to (0,0)$.