Partial Derivatives

Partial derivatives are the coordinate-level engine behind essentially all of multivariable analysis and its applications. In pure mathematics, they are the entries of the Jacobian Matrix and the components of the Gradient vector, and they appear as the coefficients in the linear approximation used to define total differentiability. In physics, partial derivatives express how a scalar field (temperature, pressure, potential energy) changes along each spatial or temporal axis while other variables are held fixed — this is precisely the meaning of thermodynamic quantities like $\left(\partial P/\partial V\right)_T$, where the subscript $T$ explicitly signals which variables are frozen. Partial differential equations (the heat equation, wave equation, Laplace's equation, Maxwell's equations, Schrödinger's equation) are, by definition, equations relating partial derivatives of an unknown function of several variables. In optimization and machine learning, the gradient descent algorithm that trains neural networks computes partial derivatives of a loss function with respect to each parameter (via backpropagation, itself an efficient application of the multivariable chain rule built from partials), and the sensitivity of a model's output to a single input feature is literally a partial derivative. In economics, partial derivatives measure marginal quantities — marginal utility, marginal cost with respect to one input — holding other inputs fixed. In statistics, partial derivatives of a likelihood function with respect to each parameter, set to zero, give the maximum-likelihood equations; in numerical analysis, finite-difference approximations to PDEs are built directly by discretizing the partial-derivative limit definition given here.

Ask the tutor: Why does continuity of the partial derivatives themselves rescue us from the pathology, and how does that lead into the definition of total differentiability?

The definition says something very concrete: to compute $\partial f/\partial x$ at a point, you slice the domain with the line through that point parallel to the $x$-axis, restrict $f$ to that line, and differentiate the resulting one-variable function in the ordinary sense. All other coordinates are treated as frozen constants during the computation. This is why, in practice, computing $\partial f/\partial x_i$ is mechanically identical to single-variable differentiation: you literally apply the same rules (power rule, product rule, chain rule) to the expression for $f$, treating every symbol other than $x_i$ as a constant. The geometric picture for $n = 2$ is the most useful mental image. If $f(x,y)$ is a surface $z = f(x,y)$ in $\mathbb{R}^3$, then fixing $y = b$ carves out a curve — the intersection of the surface with the vertical plane $y = b$ — and $\partial f/\partial x(a,b)$ is the slope of the tangent line to that curve at $x = a$. Symmetrically, $\partial f/\partial y(a,b)$ is the slope of the tangent line to the curve cut out by the plane $x = a$. These are two different tangent lines to the surface at the point $(a,b,f(a,b))$, and crucially they are only two directions among infinitely many along which one could approach the point — which is exactly why partial derivatives, by themselves, cannot see everything about the local behavior of $f$.

If $\partial f/\partial x_i$ exists on an open set $V$, we may ask whether its partial derivative with respect to some $x_j$ exists at a point of $V$. If so, we call the result a second-order partial derivative, written

Note the notational convention: in $\partial^2 f/(\partial x_j \partial x_i)$ the rightmost operator $\partial/\partial x_i$ is applied first (reading right to left), whereas in subscript notation $f_{x_i x_j}$ the leftmost subscript is applied first (reading left to right). Both conventions are standard; one must fix and respect one of them. When $i = j$ we write $\partial^2 f/\partial x_i^2$ or $f_{x_i x_i}$. When $i \neq j$, the resulting derivative is called a mixed partial derivative. This process iterates: an order-$k$ partial derivative is obtained by $k$ successive partial differentiations, in some specified order of variables, and is denoted

A priori, nothing forces $f_{xy} = f_{yx}$ at a given point — the two mixed partials are computed via different iterated limit processes (differentiate in $x$ then in $y$, versus $y$ then $x$), and different limiting procedures need not agree. It is a genuine theorem — Clairaut's theorem (also called Schwarz's theorem), covered separately — that mixed partials do coincide provided both are continuous near the point in question. Without such a continuity hypothesis, equality can fail, as the classical counterexample $f(x,y) = xy(x^2-y^2)/(x^2+y^2)$ (with $f(0,0)=0$) demonstrates: for that function $f_{xy}(0,0) = 1 \ne -1 = f_{yx}(0,0)$. We flag this now and defer the full proof and analysis to Clairaut's Theorem.

Example 1 (computing partials directly from the limit definition, and via rules, for an asymmetric function). Let $f(x,y) = x^2 y^3 + \sin(xy)$, and compute $f_x(2,1)$ and $f_y(2,1)$ two ways: by the rules, and by the raw limit definition, to confirm they agree. Via differentiation rules. Treating $y$ as constant and differentiating in $x$:

Treating $x$ as constant and differentiating in $y$:

Via the limit definition, for $f_x(2,1)$ as a check. We must show

Compute $f(2+h,1) = (2+h)^2(1)^3 + \sin\big((2+h)(1)\big) = (2+h)^2 + \sin(2+h)$, and $f(2,1) = 4 + \sin 2$. So

As $h \to 0$, the first term $\to 4$. The second term is exactly the difference quotient for the single-variable derivative of $\sin$ at $2$, so by the known fact $(\sin t)' = \cos t$, it tends to $\cos 2$. Hence the whole expression tends to $4 + \cos 2$, matching the rule-based computation exactly. This illustrates why the "rules" method is legitimate: it is nothing more than the single-variable differentiation rules (product rule, chain rule) applied to the one-variable function $t \mapsto f(t,1)$, which is precisely what the partial-derivative limit computes.

Example 2 (a function requiring the definition at a boundary/special point, and second-order mixed partials). Let

We compute $f_x(0,0)$, $f_y(0,0)$ from the definition, then compute $f_{xy}(0,0)$ and $f_{yx}(0,0)$ to see whether they agree for this particular function (they will, since it is smooth away from the origin and one can check the hypotheses of Clairaut's theorem hold — but we verify directly at the origin using limits, as good practice for the definition). First partials at the origin. Set $g(t) = f(t,0) = \dfrac{t^3 - 0}{t^2}(t) = \dfrac{t^3}{t^2}\cdot t = t^2$ for $t \ne 0$ — wait, let's compute carefully: $f(t,0) = \dfrac{t^3 - 0^3}{t^2+0^2}(t + 0) = \dfrac{t^3}{t^2}\cdot t = t \cdot t = t^2$ for $t \neq 0$, and $f(0,0) = 0 = 0^2$, so $g(t) = t^2$ for all $t$. Then

Similarly $h(t) = f(0,t) = \dfrac{0 - t^3}{t^2}(2t) = \dfrac{-t^3}{t^2}(2t) = -2t^2$ for $t\ne0$, and $h(0)=0=-2(0)^2$, so $h(t) = -2t^2$ identically, giving

Partials away from the origin, to set up the mixed partial. For $(x,y) \ne (0,0)$, write $N = (x^3-y^3)(x+2y) = x^4 + 2x^3y - xy^3 - 2y^4$ and $D = x^2+y^2$, so $f = N/D$. By the quotient rule,

We want $f_x(0,y)$ for $y \ne 0$ (needed to differentiate $f_x$ in $y$ at the origin along the $y$-axis). At points $(0,y)$ with $y \ne 0$: $N(0,y) = -2y^4$, $D(0,y) = y^2$, $N_x(0,y) = -y^3$, $D_x(0,y) = 0$. So

Together with $f_x(0,0) = 0$ computed above, we get $f_x(0,y) = -y$ for all $y$ (including $y=0$). Hence

Now the other order: by symmetry of the quotient rule, $f_y = \dfrac{N_y D - N D_y}{D^2}$ with $N_y = 2x^3 - 3xy^2 - 8y^3$, $D_y = 2y$. At points $(x,0)$ with $x \ne 0$: $N(x,0) = x^4$, $D(x,0)=x^2$, $N_y(x,0) = 2x^3$, $D_y(x,0)=0$, so

and together with $f_y(0,0)=0$ this gives $f_y(x,0) = 2x$ for all $x$. Hence

Conclusion of Example 2. We find $f_{xy}(0,0) = -1$ while $f_{yx}(0,0) = 2$: the mixed partials disagree at the origin for this function. This is not a contradiction of any theorem — it simply shows that $f_{xy}$ and $f_{yx}$ need not be continuous near $(0,0)$ here, so the hypothesis of Clairaut's theorem fails and no equality is guaranteed. This example (structurally similar to the standard textbook counterexample for Clairaut's theorem) reinforces that mixed-partial equality is a genuine theorem requiring a continuity hypothesis, not an automatic algebraic identity — the full statement and proof are developed in Clairaut's Theorem.