1.13: Functions of Several Variables

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For simplicity, we consider a function \(f = f(x, y)\) of two variables, though the results are easily generalized. The partial derivative of \(f\) with respect to \(x\) is defined as \[\frac{\partial f}{\partial x}=\underset{h\to 0}{\lim}\frac{f(x+h,y)-f(x,y)}{h},\nonumber\] and similarly for the partial derivative of \(f\) with respect to \(y\). To take the partial derivative of \(f\) with respect to \(x\), say, take the derivative of \(f\) with respect to \(x\) holding \(y\) fixed. As an example, consider \[f(x,y)=2x^3y^2+y^3.\nonumber\]

We have \[\frac{\partial f}{\partial x}=6x^2y^2,\quad\frac{\partial f}{\partial y}=4x^3y+3y^2.\nonumber\]

Second derivatives are defined as the derivatives of the first derivatives, so we have \[\frac{\partial ^2f}{\partial x^2}=12xy^2,\quad \frac{\partial ^2f}{\partial y^2}=4x^3+6y;\nonumber\] and the mixed second partial derivatives are \[\frac{\partial ^2f}{\partial x\partial y}=12x^2y,\quad\frac{\partial ^2f}{\partial y\partial x}=12x^2y.\nonumber\]

In general, mixed partial derivatives are independent of the order in which the derivatives are taken.

Partial derivatives are necessary for applying the chain rule. Consider \[df=f(x+dx, y+dy)-f(x,y).\nonumber\]

We can write \(df\) as \[\begin{aligned} df&= [ f(x + dx, y + dy) − f(x, y + dy)] + [ f(x, y + dy) − f(x, y)] \\ &=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy.\end{aligned}\]

If one has \(f = f(x(t), y(t)),\) say, then \[\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}.\nonumber\]

And if one has \(f = f(x(r, θ), y(r, θ)),\) say, then \[\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r},\quad\frac{\partial f}{\partial\theta}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial\theta}.\nonumber\]

A Taylor series of a function of several variables can also be developed. Here, all partial derivatives of \(f(x, y)\) at \((a, b)\) match all the partial derivatives of the power series. With the notation \[f_x=\frac{\partial f}{\partial x},\quad f_y=\frac{\partial f}{\partial y},\quad f_{xx}=\frac{\partial ^2f}{\partial x^2},\quad f_{xy}=\frac{\partial ^2f}{\partial x\partial y},\quad f_{yy}=\frac{\partial ^2f}{\partial y^2},\quad\text{etc.,}\nonumber\] we have \[\begin{aligned} f(x,y)&=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b) \\ &+\frac{1}{2!}\left( f_{xx}(a,b)(x-a)^2+2f_{xy}(a,b)(x-a)(y-b)+f_{yy}(a,b)(y-b)^2\right)+\cdots\end{aligned}\]