Editing Derivative (section)

=== Partial derivatives ===
{{Main|Partial derivative}}
Functions can depend upon [[function (mathematics)#Multivariate function|more than one variable]]. A [[partial derivative]] of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in [[vector calculus]] and [[differential geometry]]. As with ordinary derivatives, multiple notations exist: the partial derivative of a function <math>f(x, y, \dots)</math> with respect to the variable <math>x</math> is variously denoted by
{{block indent | em = 1.2 | text = <math>f_x</math>, <math>f'_x</math>, <math>\partial_x f</math>, <math>\frac{\partial}{\partial x}f</math>, or <math>\frac{\partial f}{\partial x}</math>,}}
among other possibilities.{{sfnm
 | 1a1 = Stewart | 1y = 2002 | 1p = [https://archive.org/details/calculus0000stew/page/947/mode/1up 947]
 | 2a1 = Christopher | 2y = 2013 | 2p = 682
}} It can be thought of as the rate of change of the function in the <math>x</math>-direction.{{sfn|Stewart|2002|p=[https://archive.org/details/calculus0000stew/page/949 949]}} Here [[∂]] is a rounded ''d'' called the '''partial derivative symbol'''.  To distinguish it from the letter ''d'', ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".{{sfnm
 | 1a1 = Silverman | 1y = 1989 | 1p = [https://books.google.com/books?id=CQ-kqE9Yo9YC&pg=PA216 216]
 | 2a1 = Bhardwaj | 2y= 2005 | 2loc = See [https://books.google.com/books?id=qSlGMwpNueoC&pg=SA6-PA4 p. 6.4]
}} For example, let {{tmath|1= f(x,y) = x^2 + xy + y^2 }}, then the partial derivative of function <math> f </math> with respect to both variables <math> x </math> and <math> y </math> are, respectively:
<math display="block"> \frac{\partial f}{\partial x} = 2x + y, \qquad \frac{\partial f}{\partial y} = x + 2y.</math>
In general, the partial derivative of a function <math> f(x_1, \dots, x_n) </math> in the direction <math> x_i </math> at the point <math>(a_1, \dots, a_n) </math> is defined to be:{{sfn|Mathai|Haubold|2017|p=[https://books.google.com/books?id=v20uDwAAQBAJ&pg=PA52 52]}}
<math display="block">\frac{\partial f}{\partial x_i}(a_1,\ldots,a_n) = \lim_{h \to 0}\frac{f(a_1,\ldots,a_i+h,\ldots,a_n) - f(a_1,\ldots,a_i,\ldots,a_n)}{h}.</math>

This is fundamental for the study of the [[functions of several real variables]]. Let <math> f(x_1, \dots, x_n) </math> be such a [[real-valued function]]. If all partial derivatives <math> f </math> with respect to <math> x_j </math> are defined at the point {{tmath|1= (a_1, \dots, a_n) }}, these partial derivatives define the vector
<math display="block">\nabla f(a_1, \ldots, a_n) = \left(\frac{\partial f}{\partial x_1}(a_1, \ldots, a_n), \ldots, \frac{\partial f}{\partial x_n}(a_1, \ldots, a_n)\right),</math>
which is called the [[gradient]] of <math> f </math> at <math> a </math>. If <math> f </math> is differentiable at every point in some domain, then the gradient is a [[vector-valued function]] <math> \nabla f </math> that maps the point <math> (a_1, \dots, a_n) </math> to the vector <math> \nabla f(a_1, \dots, a_n) </math>. Consequently, the gradient determines a [[vector field]].{{sfn|Gbur|2011|pp=36–37}}