Functions of several variables: Partial derivatives
First partial derivatives
You can treat one of two variables in a function \(f(x,y)\), say \(y\), as a constant and then you get a function that has only \(x\) as an independent variable. In the surface graph of \(f\) you are then working on a coordinate curve at a constant \(y\)-value. If this function of \(x\) is neat, you can determine its derivative: this derivative is called the first partial derivative of \(f(x,y)\) with respect to \(x\). The following three notations are commonly used for the first partial derivative: \[\frac{\partial f}{\partial x}(x,y),\quad \frac{\partial }{\partial x}f(x,y)\quad \text{and}\quad f_x(x,y)\] According to the definition of the partial derivative is \[\frac{\partial f}{\partial x}(x,y)\approx \frac{f(x+{\vartriangle}x,y)-f(x,y)}{{\vartriangle}x}\quad\text{for small }{\vartriangle}x\] Likewise you get the partial derivative with respect to \(y\) when you keep \(x\) fixed and only use \(y\) as independent variable. The usual notations are then \[\frac{\partial f}{\partial y}(x,y),\quad \frac{\partial }{\partial y}f(x,y)\quad \text{and}\quad f_y(x,y)\]
Definitions and notation The first partial derivatives of the function \(f(x,y)\) are functions \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) defined by \[\begin{aligned} \frac{\partial f}{\partial x}(x,y) &=\lim_{{\vartriangle}x\rightarrow 0}\frac{f(x+{\vartriangle}x,y)-f(x,y)}{{\vartriangle}x}\\ \\ \frac{\partial f}{\partial y}(x,y) &=\lim_{{\vartriangle}y\rightarrow 0}\frac{f(x,y+{\vartriangle}y)-f(x,y)}{{\vartriangle}y}\end{aligned}\]
When we want the partial derivative of the function \(f(x,y)\) evaluated at some point \((a,b)\), then we use the following notatopm \[\frac{\partial f}{\partial x}\Biggl|_{(a,b)} \text{instead of }\frac{\partial f}{\partial x}(a,b) \qquad\text{and}\qquad\frac{\partial f}{\partial y}\!\Biggl|_{(a,b)} \text{instead of }\frac{\partial f}{\partial y}(a,b)\] The most readable notation of a partial derivative evaluated in a point is \(f_x(a,b)\) and \(f_y(a,b)\).
Rule for computing partial derivatives The rules for differentiation of functions of one variable, that is to say, the constant factor rule and the sum, difference, product and quotient rule, can be applied to compute partial derivatives.
Partial derivative w.r.t. \(x\): \[\begin{aligned} \frac{\partial }{\partial x}(c\cdot f)&= c\cdot \frac{\partial f}{\partial x}\quad \text{for constant }c\\ \\ \frac{\partial }{\partial x}(f\pm g)&= \frac{\partial f}{\partial x}\pm \frac{\partial g}{\partial x}\\ \\ \frac{\partial }{\partial x}(f\cdot g) &= \frac{\partial f}{\partial x}\cdot g + f\cdot \frac{\partial g}{\partial x} \\ \\ \frac{\partial }{\partial x}\left(\frac{f}{g}\right) &= \frac{\displaystyle\frac{\partial f}{\partial x}\cdot g - f\cdot \frac{\partial g}{\partial x}}{g^2}\end{aligned}\]
Partial derivative w.r.t. \(y\): \[\begin{aligned} \frac{\partial }{\partial y}(c\cdot f)&= c\cdot \frac{\partial f}{\partial y}\quad \text{for constant }c\\ \\ \frac{\partial }{\partial y}(f\pm g)&= \frac{\partial f}{\partial y}\pm \frac{\partial g}{\partial y}\\ \\\frac{\partial }{\partial y}(f\cdot g) &= \frac{\partial f}{\partial y}\cdot g + f\cdot \frac{\partial g}{\partial y} \\ \\ \frac{\partial }{\partial y}\left(\frac{f}{g}\right) &= \frac{\displaystyle\frac{\partial f}{\partial y}\cdot g - f\cdot \frac{\partial g}{\partial y}}{g^2}\end{aligned}\]
The remaining part of the formula that depends on \(y\) is then \(y^4\).
The derivative (w.r.t. \(y\) ) is \(4 \cdot y^{4-1}=4 y^3\).
The final result is the product of the two intermediate results: \[\frac{\partial}{\partial y} (6x^{3}y^{4})=6x^{3}\cdot 4 y^3=24x^3y^3\]
Partial derivatives can be defined in a similar way for functions of more than two variables. For example, for the function \[f(x,y,z)=\sqrt{x^2+y^2+z^2}\] we get \[\frac{\partial}{\partial z}f(x,y,z) = \frac{z}{\sqrt{x^2+y^2+z^2}}\]
