§ 01The Big Idea — What Is a Partial Derivative?
A function of two variables f(x, y) lives on a surface in three dimensions. A partial derivative measures the slope of that surface in one specific direction while everything else stays frozen.
In single-variable calculus, the derivative f′(x) tells you the instantaneous rate of change of f as x varies. When we have a function of two variables, f(x, y), the function has infinitely many directions in which it can change. Rather than try to capture all of them at once, we pick one variable and ask: “how does f change as x increases, with y held perfectly still?” The answer is the partial derivative with respect to x, written ∂f/∂x.
The formal definition mirrors the single-variable limit definition of a derivative — the only difference is that the other variable is treated as a constant throughout:
In practice, you never have to use these limit definitions directly. Computing a partial derivative is identical to computing an ordinary derivative, with one simple rule: when differentiating with respect to x, treat every occurrence of y as if it were a numeric constant (and vice versa).
The Geometric Picture
Picture a hilly surface z = f(x, y). If you stand at the point (a, b, f(a,b)) and walk in the x-direction only (keeping y fixed at b), you trace a curve. The slope of that curve at your feet is exactly ∂f/∂x evaluated at (a, b). Likewise, walking in the y-direction gives the slope ∂f/∂y. Together, these two slopes determine the orientation of the tangent plane to the surface at that point — one of the most important geometric objects in multivariable calculus.
§ 02Notation — All the Ways to Write a Partial Derivative
Partial derivatives appear in several equivalent notations. You will see all of them in textbooks and need to be comfortable with each.
| Notation | Reads as | Notes |
|---|---|---|
| ∂f/∂x | “partial f, partial x” | Most common. The symbol ∂ (a rounded d) flags a partial derivative. |
| fx | “f sub x” | Compact subscript notation. Very common in analysis and PDE texts. |
| ∂f/∂y | “partial f, partial y” | The partial with respect to y — x treated as constant. |
| fy | “f sub y” | Subscript notation for the y partial. |
| ∂²f/∂x² | “second partial in x” | Differentiate twice with respect to x. Also written fxx. |
| ∂²f/∂y∂x | “mixed partial, y then x” | Differentiate first w.r.t. x, then w.r.t. y. Also written fxy. |
§ 03Computing First Partial Derivatives — 8 Worked Examples
The only skill required is treating the other variable as a constant. Every differentiation rule you know — power rule, product rule, chain rule — applies unchanged.
d/dx(x³y) = 3x²y, d/dx(2xy²) = 2y², d/dx(−5y³) = 0
fx = 3x²y + 2y²
d/dy(x³y) = x³, d/dy(2xy²) = 4xy, d/dy(−5y³) = −15y²
fy = x³ + 4xy − 15y²
d/dx(exy) = y exy (chain rule, inner derivative is y)
∂f/∂x = y exysin(y)
= (d/dy exy)sin(y) + exy(d/dy sin(y))
= x exysin(y) + exycos(y)
∂f/∂y = exy(x sin(y) + cos(y))
∂u/∂x = 2x
fx = 2x / (x²+y²)
fy = 2y / (x²+y²)
u′ = 2x, v′ = y (w.r.t. x)
∂f/∂x = (2x(xy+2) − (x²+1)y) / (xy+2)²
= (2x²y + 4x − x²y − y) / (xy+2)² = (x²y + 4x − y) / (xy+2)²
∂f/∂y = (x²+1) · (−1)(xy+2)−2 · x
= −x(x²+1) / (xy+2)²
§ 04Higher-Order Partial Derivatives — Second and Mixed Partials
Just as we can differentiate f′(x) again to get f″(x), we can differentiate partial derivatives to get second-order partials. A function of two variables has four second-order partials.
fyy = ∂²f/∂y² (differentiate fy again w.r.t. y)
fxy = ∂²f/∂y∂x (differentiate fx w.r.t. y)
fyx = ∂²f/∂x∂y (differentiate fy w.r.t. x)
The first two (fxx and fyy) are called pure second partials. The last two (fxy and fyx) are called mixed partials. Each measures how the slope in one direction changes as we move in the same or a different direction.
§ 05Clairaut’s Theorem — Why Mixed Partials Are Equal
fxy(x, y) = fyx(x, y) for all (x, y) in D
In plain English: if the mixed partials are both continuous, it does not matter in which order you differentiate — you get the same answer. This is true for the overwhelming majority of functions you will encounter at undergraduate level (all polynomials, exponentials, trig functions, and their combinations).
Clairaut’s theorem is enormously useful. It means you can choose whichever order of mixed differentiation looks simpler. It also underlies the structure of Hessian matrices (used to classify critical points) and is a key hypothesis in the theory of exact differential equations.
d/dy[y cos(xy)] = cos(xy) + y(−sin(xy))·x = cos(xy) − xy sin(xy)
d/dy[2xy] = 2x
fxy = cos(xy) − xy sin(xy) + 2x
d/dx[x cos(xy)] = cos(xy) + x(−sin(xy))·y = cos(xy) − xy sin(xy)
d/dx[x²] = 2x
fyx = cos(xy) − xy sin(xy) + 2x
§ 06Geometric Meaning — Slopes on a Surface & the Tangent Plane
Partial derivatives are not just algebraic gadgets. They have a precise geometric interpretation that connects back to the slope of a tangent line — now in 3D.
Slopes of Traces
When we fix y = b and let x vary, the surface z = f(x, y) is “sliced” by the plane y = b, producing a curve called a trace. The partial derivative fx(a, b) is exactly the slope of the tangent line to this trace at the point (a, b, f(a,b)). Similarly, fy(a, b) is the slope of the trace obtained by fixing x = a.
The Tangent Plane Equation
The two trace slopes together determine the equation of the tangent plane at (a, b, f(a,b)):
This formula is the 2D analogue of the tangent line y = f(a) + f′(a)(x−a). It is the foundation of linear approximation in multivariable calculus (the linearisation of f near (a, b)), and it feeds directly into the gradient vector and directional derivatives that follow in the next topic.
fy = 2y → fy(1,2) = 4
fy = xey → fy(2,0) = 2e&sup0; = 2
§ 07Partial Derivatives of Functions of Three or More Variables
The same rule extends without change to functions of three, four, or any number of variables. When finding ∂f/∂x, treat every variable except x as a constant.
∂P/∂T = nR/V
Meaning: increasing temperature at fixed volume increases pressure proportionally.
∂P/∂V = −nRT/V²
Meaning: increasing volume at fixed temperature decreases pressure (negative derivative).
§ 08What Comes Next — Related Topics
Partial derivatives are the gateway to all of multivariable calculus. Once you are confident with ∂f/∂x and ∂f/∂y, every topic below builds directly on them.
§ 09Common Mistakes — What to Watch Out For
§ 10Test Your Knowledge — 10-Question Quiz
Select your answer and click Submit. Correct answers light up green; wrong answers reveal the right choice with a full explanation.