Multivariate functions: Partial derivatives
Partial derivatives of the first order
When dealing with a bivariate function \(f(x,y)\), you can keep fixed one of the two variables, say \(y\), and regard it as a constant. The result will be a function of a single variable \(x\). We study the derivative of this function of \(x\).
Partial derivative The (first order) partial derivatives of the bivariate function \(f(x,y)\) are the functions \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) defined by \[ \begin{array}{rcl}\displaystyle\frac{\partial f}{\partial x}\!(x,y) &=&\displaystyle\lim_{h\rightarrow 0}\frac{f(x+h,y)-f(x,y)}{h}\\ \\\displaystyle\frac{\partial f}{\partial y}\!(x,y) &=&\displaystyle\lim_{h\rightarrow 0}\frac{f(x,y+h)-f(x,y)}{h}\end {array}\]
The following four ways to write the partial derivative with respect to #x# are commonly used: \[\frac{\partial f}{\partial x}\!(x,y)\text{,}\quad \frac{\partial }{\partial x}\!f(x,y)\text{,}\quad f_x(x,y)\text{,}\quad\text{and}\quad f'_x(x,y)\] Similarly, for the partial derivative with respect to #y#: \[\frac{\partial f}{\partial y}\!(x,y)\text{,}\quad \frac{\partial }{\partial y}\!f(x,y)\text{,}\quad f_y(x,y)\text{,}\quad\text{and}\quad f'_y(x,y)\]
When we want to refer to the partial derivative of the function \(f(x,y)\) at some point \(\rv{a,b}\), we use the following notation \[\frac{\partial f}{\partial x}\!\Biggl|_{\rv{a,b}} \text{ instead of }\frac{\partial f}{\partial x}\!(a,b) \] and \[\frac{\partial f}{\partial y}\!\Biggl|_{\rv{a,b}} \text{ instead of }\frac{\partial f}{\partial y}\!(a,b)\] Other common notation in this case is \(f_x(a,b)\) and \(f_y(a,b)\).
The existence and many of the properties of the partial derivatives are not different from those of derivatives of functions of a single variable.
We formulate the definitions for the functions of two variables, but it will be clear how they can be defined for more than two variables.
The words "first order" are only important when we discuss higher partial derivatives; usually they are left out.
Why don't we write #\frac{\dd f}{\dd x}# instead of #\frac{\partial f}{\partial x}#? Because in the case where #x# and #y# are functions of two variables, say #s# and #t#, the expressions #\frac{\dd f}{\dd t}# and #\frac{\partial f}{\partial t}# have different meanings. We will come back to this later.
Calculation rules for partial derivatives
The calculation rules for derivatives of functions of a single variable are applicable for partial derivatives. In particular, for the partial derivative with respect to \(x\), we have \[ \begin{array}{rclcl} \frac{\partial }{\partial x}\!(c\cdot f)&=& c\cdot \frac{\partial f}{\partial x}\quad \text{for constant }c&\phantom{x}&\color{blue}{\text{constant factor rule}}\\ \\ \frac{\partial }{\partial x}\!(f+ g)&=& \frac{\partial f}{\partial x}+ \frac{\partial g}{\partial x}&\phantom{x}&\color{blue}{\text{sum rule}}\\ \\ \frac{\partial }{\partial x}\!(f\cdot g) &=& \frac{\partial f}{\partial x}\cdot g + f\cdot \frac{\partial g}{\partial x}&\phantom{x}&\color{blue}{\text{product rule}}\\ \\ \frac{\partial }{\partial x}\!\!\left(\frac{f}{g}\right) &=& \frac{\displaystyle\frac{\partial f}{\partial x}\cdot g - f\cdot \frac{\partial g}{\partial x}}{g^2}&\phantom{x}&\color{blue}{\text{quotient rule}}\end{array}\]
Partial derivative with respect to \(y\): same rules, with #x# replaced by #y#.
The chain rule will be discussed later.
For functions of more than two variables, the partial derivatives can be defined in a similar manner.
We consider \(y\) as a parameter and \(x\) as the independent variable. So \(5y\) is constant.
The remainder of the formula depends on \(x\): it is \(x^8\). The derivative of it with respect to \(x\) is equal to \(8 \cdot x^{8-1}=8 x^7\) .
The end result is the product of the two intermediate results: \[\frac{\partial}{\partial x}\!(5x^8\cdot y)=5y\cdot 8 x^7=40x^7\cdot y\tiny.\]
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