Partial Derivatives

Motivation. For $F(x,y)$, holding $y=y_0$ fixed turns $F(x,y_0)$ into a function of $x$ alone, whose graph is the curve cut from the surface $z=F(x,y)$ by the plane $y=y_0$; its ordinary derivative with respect to $x$, evaluated at $x=x_0$, is the slope of the tangent to that curve — this is the rate of change of $F$ in the $x$-direction only, with $y$ held fixed. Symmetrically for $y$.

Definition 8.8 (Partial Derivative). Let $A=\{(x,y)\mid a<x<b,\,c<y<d\}\subset\mathbb R^2$, $F:A\to\mathbb R$, $(x_0,y_0)\in A$.

1. $F$ has a partial derivative with respect to $x$ at $(x_0,y_0)$ if $\displaystyle\lim_{h\to0}\frac{F(x_0+h,y_0)-F(x_0,y_0)}{h}$ exists; the limit value is denoted $\dfrac{\partial F}{\partial x}(x_0,y_0)$, also written $F_x(x_0,y_0)$.
2. $F$ has a partial derivative with respect to $y$ at $(x_0,y_0)$ if $\displaystyle\lim_{k\to0}\frac{F(x_0,y_0+k)-F(x_0,y_0)}{k}$ exists; the limit value is $\dfrac{\partial F}{\partial y}(x_0,y_0)$, also written $F_y(x_0,y_0)$. Read $\partial F$ as "partial $F$", $\partial x$ as "partial $x$", and $\dfrac{\partial F}{\partial x}$ as "partial $F$ by partial $x$" (or "dho $F$ by dho $x$"). If $F$ has a partial derivative w.r.t. $x$ at every point of $A$, then $\dfrac{\partial F}{\partial x}(x,y)$ is itself a new function on $A$. Partial derivatives for three or more variables are defined exactly the same way, one variable at a time with all others frozen. Mechanically, all the usual rules of differentiation (sum, product, quotient, chain) apply unchanged — the only new bookkeeping is that every variable except the one being differentiated is treated as a constant.
   Watch out

   Unlike one variable — where differentiability always implies continuity — the existence of both $F_x$ and $F_y$ at a point does NOT guarantee $F$ is continuous there. Example: $f(x,y)=0$ if $xy\ne0$, $f(x,y)=1$ if $xy=0$. Along the $x$-axis ($y=0$) or the $y$-axis ($x=0$), $f\equiv1$, so both $f_x(0,0)=0$ and $f_y(0,0)=0$ exist; yet along the line $y=x$ with $x\ne0$, $f(x,x)=0\ne1=f(0,0)$, so $f$ is not continuous at $(0,0)$.

   Worked example. For $F(x,y)=x^3y^2+7xy$, holding $y$ fixed and differentiating w.r.t. $x$: $F_x=3x^2y^2+7y$; holding $x$ fixed and differentiating w.r.t. $y$: $F_y=2x^3y+7x$. Evaluate at any given point exactly as with a one-variable derivative. Second-order and mixed partial derivatives. Since $F_x$ is again a function of $(x,y)$, it can be partially differentiated again:

   $\frac{\partial^2F}{\partial x^2}=F_{xx}=\frac{\partial}{\partial x}\!\left(\frac{\partial F}{\partial x}\right), \quad \frac{\partial^2F}{\partial y^2}=F_{yy}, \quad \frac{\partial^2F}{\partial y\,\partial x}=F_{xy}=\frac{\partial}{\partial y}\!\left(\frac{\partial F}{\partial x}\right), \quad \frac{\partial^2F}{\partial x\,\partial y}=F_{yx}=\frac{\partial}{\partial x}\!\left(\frac{\partial F}{\partial y}\right).$

   Higher (third and beyond) mixed partials continue the same pattern, one variable at a time, e.g. $\dfrac{\partial^3F}{\partial y\,\partial x\,\partial y}=\dfrac{\partial}{\partial y}\!\left(\dfrac{\partial^2F}{\partial x\,\partial y}\right)$. …