= Partial derivative
{title2=$\partial$}
{wiki}
= Partial differentiation
{synonym}
A <partial derivative> measures the local response of a <function> to one input while its other inputs stay fixed. For a <function> $f$ on an <open subset> of $\mathbb R^2$, its <partial derivatives> at $(a,b)$ are the <limits>
$$
f_x(a,b)=\lim_{h\to0}\frac{f(a+h,b)-f(a,b)}{h},
\qquad
f_y(a,b)=\lim_{k\to0}\frac{f(a,b+k)-f(a,b)}{k},
$$
when the respective <limits> exist. Each is an ordinary <derivative> of a one-variable slice: $x\mapsto f(x,b)$ or $y\mapsto f(a,y)$. This is why the other variable is treated as a constant during <partial differentiation>.<Calculus Volume 3 by Gilbert Strang and others/4.3 Partial Derivatives>[Calculus Volume 3, §4.3, definitions (4.12)–(4.13) and the one-variable slice interpretation]{ref}
For an original example, take the <polynomial function>
$$
f(x,y)=x^2+xy+2y^2.
$$
Holding $y$ fixed gives $f_x=2x+y$, while holding $x$ fixed gives $f_y=x+4y$. At $(1,1)$, the height is $f=4$ and the two <partial derivatives> are \b[$f_x=3$ and $f_y=5$]. The values answer different experiments on the same mathematical model: change $x$ alone, or change $y$ alone.
The two exact slices through that point are
$$
f(1+t,1)=4+3t+t^2,\qquad
f(1,1+t)=4+5t+2t^2.
$$
Dividing either height change by $t\ne0$ gives $3+t$ or $5+2t$; taking the <limit> $t\to0$ recovers the two slopes. At $t=0.1$, the finite height changes are $0.31$ and $0.52$, although their <linear approximations> are $0.3$ and $0.5$. A <derivative> measures the limiting response, rather than an exact finite change.
\Image[/partial-derivative-coordinate-slices.png]
{title=Two coordinate slices through one point give different <partial derivatives>}
{description=Original <contour plot> of $f=x^2+xy+2y^2$, with the point $(1,1)$ and coordinate paths. The right panel compares the exact slice height changes $3t+t^2$ and $5t+2t^2$ with their <tangent line> predictions $3t$ and $5t$. Both slices pass through height $4$.}
For a <function> of $n$ variables, $\partial_j f(a)$ varies only its $j$th coordinate: it is the same <difference quotient> along $e_j$, the $j$th <standard basis> <vector>. This gives a practical <sensitivity analysis>: one can compare how a <mathematical model> responds to changes in its individual inputs, provided the chosen <system of units> and the quantities held fixed are stated. Comparing the numerical sizes of two <partial derivatives> in different units does not by itself identify a physically preferable input.
Existence of the <partial derivatives> at one point is weaker than <differentiability>. The example under <partial derivatives do not imply continuity> shows how both coordinate tests can miss a bad diagonal approach. When the first <partial derivatives> are continuous in a neighborhood, they do give <differentiability> there, allowing the <total differential> to combine all small input changes.
Back to article page
