In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
The partial derivative of a function {\displaystyle f(x,y,\dots )}{\displaystyle f(x,y,\dots )} with respect to the variable {\displaystyle x}x is variously denoted by
{\displaystyle f_{x}}f_{x},{\displaystyle f’{x}}{\displaystyle f’{x}}, {\displaystyle \partial {x}f}{\displaystyle \partial {x}f}, {\displaystyle \ D{x}f}{\displaystyle \ D{x}f}, {\displaystyle D_{1}f}{\displaystyle D_{1}f}, {\displaystyle {\frac {\partial }{\partial x}}f}{\displaystyle {\frac {\partial }{\partial x}}f}, or {\displaystyle {\frac {\partial f}{\partial x}}}\frac{\partial f}{\partial x}.
It can be thought of as the rate of change of the function in the {\displaystyle x}x-direction.
Sometimes, for {\displaystyle z=f(x,y,\ldots )}{\displaystyle z=f(x,y,\ldots )}, the partial derivative of {\displaystyle z}z with respect to {\displaystyle x}x is denoted as {\displaystyle {\tfrac {\partial z}{\partial x}}.}{\displaystyle {\tfrac {\partial z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:
{\displaystyle f’{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).}{\displaystyle f’{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).}
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841.[1]
Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n} and {\displaystyle f:U\to \mathbb {R} }{\displaystyle f:U\to \mathbb {R} } a function. The partial derivative of f at the point {\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{n})\in U}{\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{n})\in U} with respect to the i-th variable xi is defined as
{\displaystyle {\begin{aligned}{\frac {\partial }{\partial x_{i}}}f(\mathbf {a} )&=\lim {h\to 0}{\frac {f(a{1},\ldots ,a_{i-1},a_{i}+h,a_{i+1},\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\dots ,a_{n})}{h}}\&=\lim {h\to 0}{\frac {f(\mathbf {a} +h\mathbf {e{i}} )-f(\mathbf {a} )}{h}}\end{aligned}}}{\displaystyle {\begin{aligned}{\frac {\partial }{\partial x_{i}}}f(\mathbf {a} )&=\lim {h\to 0}{\frac {f(a{1},\ldots ,a_{i-1},a_{i}+h,a_{i+1},\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\dots ,a_{n})}{h}}\&=\lim {h\to 0}{\frac {f(\mathbf {a} +h\mathbf {e{i}} )-f(\mathbf {a} )}{h}}\end{aligned}}}
Even if all partial derivatives ∂f/∂xi(a) exist at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C1 function. This can be used to generalize for vector valued functions, {\displaystyle f:U\to \mathbb {R} ^{m}}{\displaystyle f:U\to \mathbb {R} ^{m}}, by carefully using a componentwise argument.
The partial derivative {\displaystyle {\frac {\partial f}{\partial x}}}\frac{\partial f}{\partial x} can be seen as another function defined on U and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut’s theorem:
{\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}.}{\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}.}
Further information: ∂
For the following examples, let {\displaystyle f}f be a function in {\displaystyle x,y}x,y and {\displaystyle z}z.
First-order partial derivatives:
{\displaystyle {\frac {\partial f}{\partial x}}=f’_{x}=\partial {x}f.}{\displaystyle {\frac {\partial f}{\partial x}}=f’{x}=\partial _{x}f.}
Second-order partial derivatives:
{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}=f’'_{xx}=\partial _{xx}f=\partial {x}^{2}f.}{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}=f’'{xx}=\partial _{xx}f=\partial _{x}^{2}f.}
Second-order mixed derivatives:
{\displaystyle {\frac {\partial ^{2}f}{\partial y,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=(f’{x})'{y}=f’'_{xy}=\partial {yx}f=\partial {y}\partial {x}f.}{\displaystyle {\frac {\partial ^{2}f}{\partial y,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=(f’{x})'{y}=f’'{xy}=\partial _{yx}f=\partial _{y}\partial _{x}f.}
Higher-order partial and mixed derivatives:
{\displaystyle {\frac {\partial ^{i+j+k}f}{\partial x^{i}\partial y^{j}\partial z{k}}}=f{(i,j,k)}=\partial _{x}^{i}\partial _{y}^{j}\partial _{z}^{k}f.}{\displaystyle {\frac {\partial ^{i+j+k}f}{\partial x^{i}\partial y^{j}\partial z{k}}}=f{(i,j,k)}=\partial _{x}^{i}\partial _{y}^{j}\partial _{z}^{k}f.}
When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics, the partial derivative of {\displaystyle f}f with respect to {\displaystyle x}x, holding {\displaystyle y}y and {\displaystyle z}z constant, is often expressed as
{\displaystyle \left({\frac {\partial f}{\partial x}}\right){y,z}.}{\displaystyle \left({\frac {\partial f}{\partial x}}\right){y,z}.}
Conventionally, for clarity and simplicity of notation, the partial derivative function and the value of the function at a specific point are conflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like
{\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}}{\frac {\partial f(x,y,z)}{\partial x}}
is used for the function, while
{\displaystyle {\frac {\partial f(u,v,w)}{\partial u}}}{\displaystyle {\frac {\partial f(u,v,w)}{\partial u}}}
might be used for the value of the function at the point {\displaystyle (x,y,z)=(u,v,w)}{\displaystyle (x,y,z)=(u,v,w)}. However, this convention breaks down when we want to evaluate the partial derivative at a point like {\displaystyle (x,y,z)=(17,u+v,v^{2})}{\displaystyle (x,y,z)=(17,u+v,v^{2})}. In such a case, evaluation of the function must be expressed in an unwieldy manner as
{\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}(17,u+v,v^{2})}{\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}(17,u+v,v^{2})}
or
{\displaystyle \left.{\frac {\partial f(x,y,z)}{\partial x}}\right|{(x,y,z)=(17,u+v,v^{2})}}{\displaystyle \left.{\frac {\partial f(x,y,z)}{\partial x}}\right|{(x,y,z)=(17,u+v,v^{2})}}
in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with {\displaystyle D_{i}}D_{i} as the partial derivative symbol with respect to the ith variable. For instance, one would write {\displaystyle D_{1}f(17,u+v,v^{2})}{\displaystyle D_{1}f(17,u+v,v^{2})} for the example described above, while the expression {\displaystyle D_{1}f}{\displaystyle D_{1}f} represents the partial derivative function with respect to the 1st variable.[2]
For higher order partial derivatives, the partial derivative (function) of {\displaystyle D_{i}f}{\displaystyle D_{i}f} with respect to the jth variable is denoted {\displaystyle D_{j}(D_{i}f)=D_{i,j}f}{\displaystyle D_{j}(D_{i}f)=D_{i,j}f}. That is, {\displaystyle D_{j}\circ D_{i}=D_{i,j}}{\displaystyle D_{j}\circ D_{i}=D_{i,j}}, so that the variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, Clairaut’s theorem implies that {\displaystyle D_{i,j}=D_{j,i}}{\displaystyle D_{i,j}=D_{j,i}} as long as comparatively mild regularity conditions on f are satisfied.
Main article: Gradient
An important example of a function of several variables is the case of a scalar-valued function f(x1, …, xn) on a domain in Euclidean space {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n} (e.g., on {\displaystyle \mathbb {R} {2}}\R2 or {\displaystyle \mathbb {R} ^{3}}\mathbb{R} ^{3}). In this case f has a partial derivative ∂f/∂xj with respect to each variable xj. At the point a, these partial derivatives define the vector
{\displaystyle \nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right).}\nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right).
This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f which takes the point a to the vector ∇f(a). Consequently, the gradient produces a vector field.
A common abuse of notation is to define the del operator (∇) as follows in three-dimensional Euclidean space {\displaystyle \mathbb {R} ^{3}}\mathbb{R} ^{3} with unit vectors {\displaystyle {\hat {\mathbf {i} }},{\hat {\mathbf {j} }},{\hat {\mathbf {k} }}}{\hat {{\mathbf {i}}}},{\hat {{\mathbf {j}}}},{\hat {{\mathbf {k}}}}:
{\displaystyle \nabla =\left[{\frac {\partial }{\partial x}}\right]{\hat {\mathbf {i} }}+\left[{\frac {\partial }{\partial y}}\right]{\hat {\mathbf {j} }}+\left[{\frac {\partial }{\partial z}}\right]{\hat {\mathbf {k} }}}{\displaystyle \nabla =\left[{\frac {\partial }{\partial x}}\right]{\hat {\mathbf {i} }}+\left[{\frac {\partial }{\partial y}}\right]{\hat {\mathbf {j} }}+\left[{\frac {\partial }{\partial z}}\right]{\hat {\mathbf {k} }}}
Or, more generally, for n-dimensional Euclidean space {\displaystyle \mathbb {R} ^{n}}\mathbb {R} ^{n} with coordinates {\displaystyle x_{1},\ldots ,x_{n}}x_{1},\ldots ,x_{n} and unit vectors {\displaystyle {\hat {\mathbf {e} }}{1},\ldots ,{\hat {\mathbf {e} }}{n}}{\displaystyle {\hat {\mathbf {e} }}{1},\ldots ,{\hat {\mathbf {e} }}{n}}:
{\displaystyle \nabla =\sum {j=1}^{n}\left[{\frac {\partial }{\partial x{j}}}\right]{\hat {\mathbf {e} }}{j}=\left[{\frac {\partial }{\partial x{1}}}\right]{\hat {\mathbf {e} }}{1}+\left[{\frac {\partial }{\partial x{2}}}\right]{\hat {\mathbf {e} }}{2}+\dots +\left[{\frac {\partial }{\partial x{n}}}\right]{\hat {\mathbf {e} }}{n}}{\displaystyle \nabla =\sum {j=1}^{n}\left[{\frac {\partial }{\partial x{j}}}\right]{\hat {\mathbf {e} }}{j}=\left[{\frac {\partial }{\partial x_{1}}}\right]{\hat {\mathbf {e} }}{1}+\left[{\frac {\partial }{\partial x{2}}}\right]{\hat {\mathbf {e} }}{2}+\dots +\left[{\frac {\partial }{\partial x{n}}}\right]{\hat {\mathbf {e} }}_{n}}