Partial derivatives, the total differential and the multivariable chain rule

A partial derivative of a multi-variable function is its derivative with respect to one of those variables, with the other variables held constant. For example, the partial derivatives of $z=f(x,y)$ with respect to $x$ and $y$ are defined as

$\left(\frac{\partial z}{\partial x}\right)_y=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}\;\;\;\;\;\;\;\;(12)$

$\left(\frac{\partial z}{\partial y}\right)_x=\lim_{\Delta y\rightarrow 0}\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}\;\;\;\;\;\;\;\;(13)$

respectively, where the symbol $(\;\;)_i$ means that the variable $i$ is held constant (the symbol may be omitted for simplicity).

If $z=x^3y^2+e^{xy}$ , then $\left(\frac{\partial z}{\partial x}\right)_y=3x^2y^2+ye^{xy}$ .

The total differential of a multi-variable function is its change with respect to the changes in all the independent variables. For example, the total differential of the function $z=f(x,y)$ is

$dz=\left(\frac{\partial z}{\partial x}\right)_ydx+\left(\frac{\partial z}{\partial y}\right)_xdy\;\;\;\;\;\;\;\;(14)$

Question

How is eq14 derived?

Answer

The total change in $z$ is $\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)$ , which is equivalent to

$\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)+f(x,y+\Delta y)-f(x,y)$

Multiplying the 1^st and 2^nd terms on the RHS of the above equation by $\frac{\Delta x}{\Delta x}$ and the 3^rd and 4^th terms by $\frac{\Delta y}{\Delta y}$ ,

$\Delta z=\left[\frac{f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)}{\Delta x}\right]\Delta x+\left[\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}\right]\Delta y\;\;\;\;(14a)$

Taking the limits $\Delta x\rightarrow 0$ and $\Delta y\rightarrow 0$

$dz=\lim_{\Delta x\rightarrow 0}\left[\frac{f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)}{\Delta x}\right]\Delta x+\lim_{\Delta y\rightarrow 0}\left[\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}\right]\Delta y$

Since the 1^st term on the RHS of the above equation is with respect to a change in $x$ , $y$ is a constant with $y+\Delta y\equiv y$ and

$dz=\lim_{\Delta x\rightarrow 0}\left[\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}\right]\Delta x+\lim_{\Delta y\rightarrow 0}\left[\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}\right]\Delta y$

Substituting eq12 and eq13 in the above equation, we have eq14.

In general, the total differential of the function $z=f(x_1,x_2,\cdots,x_n)$ is

$dz=\left(\frac{\partial z}{\partial x_1}\right)_{x_2,x_3,\cdots,x_n}dx_1+\left(\frac{\partial z}{\partial x_2}\right)_{x_1,x_3,\cdots,x_n}dx_2+\cdots+\left(\frac{\partial z}{\partial x_n}\right)_{x_1,x_2,\cdots,x_{n-1}}dx_n$

If the variables themselves depend on another variable $t$ , i.e. $x=x(t)$ and $y=y(t)$ , we divide eq14a throughout by $\Delta t$ to give

$\frac{\Delta z}{\Delta t}=\left[\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}\right]\frac{\Delta x}{\Delta t}+\left[\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}\right]\frac{\Delta y}{\Delta t}$

Since $\Delta x\rightarrow 0$ and $\Delta y\rightarrow 0$ as $\Delta t\rightarrow 0$ , if we take the limit $\Delta t\rightarrow 0$ , we have

$\frac{dz}{dt}=\left(\frac{\partial z}{\partial x}\right)_y \frac{dx}{dt}+\left(\frac{\partial z}{\partial y}\right)_x\frac{dy}{dt}\;\;\;\;\;\;\;\;(14b)$

Eq14b is known as the multivariable chain rule, which is also known as the total derivative of $z$ .

Next, we shall derive some of useful identities. With respect to eq14, if $y$ is a constant, $(dz)_y=\left(\frac{\partial z}{\partial x}\right)_y=(dx)_y$ , which when divided throughout by $(dz)_y$ gives $1=\left(\frac{\partial z}{\partial x}\right)_y\left(\frac{\partial x}{\partial z}\right)_y$ or

$\left(\frac{\partial z}{\partial x}\right)_y=\frac{1}{\left(\frac{\partial x}{\partial z}\right)_y}\;\;\;\;\;\;\;\;(15)$

If $z$ is a constant, eq14 becomes $0=\left(\frac{\partial z}{\partial x}\right)_y(dx)_z+\left(\frac{\partial z}{\partial y}\right)_x(dy)_z$ , which when divided by $(dz)_z$ gives $0=\left(\frac{\partial z}{\partial x}\right)_y\left(\frac{\partial x}{\partial y}\right)_z+\left(\frac{\partial z}{\partial y}\right)_x$ . Using the reciprocal identity of eq15, we have

$\left(\frac{\partial y}{\partial z}\right)_x\left(\frac{\partial z}{\partial x}\right)_y\left(\frac{\partial x}{\partial y}\right)_z=-1\;\;\;\;\;\;\;\;(16)$

If $dy=0$ in eq14b, we have the chain rule:

$\frac{dz}{dt}=\frac{dz}{dx}\frac{dx}{dt}\;\;\;\;\;\;\;\;(17)$

Finally, the second partial derivative of $z$ is defined as $\frac{\partial^2z}{\partial x\partial y}=\left[\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)_x\right]_y$ .

Partial derivatives, the total differential and the multivariable chain rule

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