Step By Step Calculus » 11.2 - Definition of Partial Derivative

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Synopsis
  • The functions \displaystyle \frac{\partial f}{\partial x}\displaystyle \frac{\partial f}{\partial x} and \displaystyle \frac{\partial f}{\partial y}\displaystyle \frac{\partial f}{\partial y} defined by
    \displaystyle \frac{\partial f}{\partial x}(x,y)=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h} \quad\displaystyle \frac{\partial f}{\partial x}(x,y)=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h} \quadand\displaystyle \quad \frac{\partial f}{\partial y}(x,y)=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h} \displaystyle \quad \frac{\partial f}{\partial y}(x,y)=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}
    for each (x, y)\in D_f(x, y)\in D_f where the limit exists are called the partial derivatives of ff with respect to xx and yy respectively.
  • The function f(x,y)f(x,y) has a limit LL as (x,y)(x,y) approaches (a,b)(a,b) if for every \epsilon>0\epsilon>0 there exists a \delta>0\delta>0 such that
    \displaystyle |f(x,y)-L|<\epsilon \quad\displaystyle |f(x,y)-L|<\epsilon \quadfor all \displaystyle (x,y)\in D_f\displaystyle (x,y)\in D_f whenever \displaystyle |x-a|\vee|y-b|<\delta, \displaystyle |x-a|\vee|y-b|<\delta,
    where “m\vee nm\vee n” means the maximum of mm and nn.
  • A function f(x,y)f(x,y) is said to be continuous at (a,b)(a,b) if
    \displaystyle \lim_{(x,y)\to(a,b)}=f(a,b).
    \displaystyle \lim_{(x,y)\to(a,b)}=f(a,b).
    f(x,y)f(x,y) is said to be continuous if it is continuous at every point of its domain.
  • For a two variable function u=f(x,y)u=f(x,y), the existence of partial derivatives at a particular point (a,b)(a,b) is not enough for the function to be continuous at (a,b)(a,b). It should also satisfy
    \displaystyle \Delta u= f_x(a,b)\Delta x+f_y(a,b)\Delta y+\epsilon_1\Delta x+\epsilon_2\Delta y,
    \displaystyle \Delta u= f_x(a,b)\Delta x+f_y(a,b)\Delta y+\epsilon_1\Delta x+\epsilon_2\Delta y,
    where \epsilon_1\epsilon_1 and \epsilon_2\epsilon_2 go to 00 as both \Delta x, \Delta y\to 0\Delta x, \Delta y\to 0.