Difficulty: Intermediate | Prerequisites: Comfort with single-variable differentiation (product rule, quotient rule, chain rule) and familiarity with multivariable functions.
Partial derivatives extend the idea of ordinary differentiation to functions of two or more variables. Instead of asking "how fast is f changing?", you ask "how fast is f changing if I nudge just one variable and hold the others fixed?" This sits at the heart of multivariable calculus and feeds directly into gradients, directional derivatives, and optimisation. You need solid single-variable differentiation (product rule, quotient rule, chain rule) before any of this will land.
A partial derivative measures the rate of change of a multivariable function with respect to one variable while all other variables are treated as constants. You compute them using the same differentiation rules you already know (product, quotient, chain), just applied one variable at a time. The sign of a partial derivative tells you whether the function is increasing or decreasing in that direction.
Partial derivative (f_x or ∂f/∂x)
The derivative of f(x, y) with respect to x, computed by treating y as a constant. In simple terms, it is the slope of the surface in the x-direction at a given point.
Partial derivative (f_y or ∂f/∂y)
The derivative of f(x, y) with respect to y, computed by treating x as a constant. Think of it as the slope of the surface in the y-direction.
Mixed partial derivative (f_xy or f_yx)
A second-order partial derivative where you differentiate with respect to one variable, then the other. For most well-behaved functions, f_xy = f_yx (Clairaut's theorem).
Contour map (level curves)
A 2D plot showing curves along which f(x, y) takes a constant value. Think of it as a topographic map: closely spaced contours mean a steep slope, and walking along a contour means no change in f.
Tangent line in the x-direction (or y-direction)
The line tangent to the surface z = f(x, y) at a point, restricted to the plane where the other variable is held fixed. Its slope equals the corresponding partial derivative.
The core technique: differentiate with respect to one variable while treating all others as constants. Every single-variable rule carries over.
Direct differentiation. Given f(x, y) = 4x² + xy − 4y² + 2x − y, treat y as a constant when finding f_x. You get f_x = 8x + y + 2. For f_y, treat x as a constant: f_y = x − 8y − 1.
Product rule. When f is a product of two expressions that both depend on x, apply the product rule in x while keeping y constant. For f(x, y) = (x² + 2y)(y² − x), we get f_x = 2x(y² − x) + (x² + 2y)(−1) = 2xy² − 2y − 3x².
Quotient rule. For f(x, y) = 4x²y / (2x² + 2y²), differentiating with respect to x requires the quotient rule. The numerator of the derivative simplifies to 16xy³ when you write it over (2x² + 2y²)².
Chain rule and logarithms. For f(x, y) = 5(y² − x²) ln(x + y), finding f_x means applying the product rule together with the chain rule for ln(x + y). At the point (0, 2) this gives a slope of 10.
Fundamental theorem of calculus. When f(x, y) = ∫ from y to x of cos(t⁶) dt, finding f_y uses the fact that ∫ from y to x = −∫ from x to y. By the fundamental theorem, f_y = −cos(y⁶).
Multivariable expressions with different letters. For u = x·e^(−6t)·sin θ, differentiating with respect to t while holding x and θ fixed gives u_t = −6x·e^(−6t)·sin θ.
Reading a contour map. If you walk in the x-direction from a point P and the contour values decrease, then f_x < 0 at P. If you walk in the y-direction and stay on the same contour (the elevation does not change), then f_y = 0 at P.
Reading a 3D surface graph. The partial derivative f_x is the slope of the tangent line to the surface in the x-direction. If the surface falls as you move in the positive x-direction, f_x < 0. Similarly for f_y in the y-direction. A point where the surface slopes downward in both the x and y directions has f_x < 0 and f_y < 0.
The key idea: partial derivatives encode directional slopes on the surface. You can read their signs from any visual representation of f without computing anything.
A mixed partial derivative like f_yx means: first differentiate with respect to y, then differentiate that result with respect to x.
Worked example. For f(x, y) = x² cos(xy):
First, f_y = −x³ sin(xy) (using the chain rule on cos(xy), with the inner derivative being x).
Then, f_yx = ∂/∂x [−x³ sin(xy)] = −3x² sin(xy) − x³ y cos(xy).
This simplifies to f_yx = −x²(3 sin(xy) + xy cos(xy)).
Clairaut's theorem guarantees that f_xy = f_yx provided the mixed partials are continuous, which they are for all the functions you will meet in this course.
Product rule (partial in x): ∂/∂x [g(x,y) · h(x,y)] = g_x · h + g · h_x
Quotient rule (partial in x): ∂/∂x [g/h] = (g_x · h − g · h_x) / h²
Chain rule: If f = g(u(x,y)), then f_x = g'(u) · u_x
Fundamental theorem of calculus link: If f(x,y) = ∫ from a(y) to b(x) of g(t) dt, then f_x = g(b(x)) · b'(x) and f_y = −g(a(y)) · a'(y)
Clairaut's theorem: If f_xy and f_yx are both continuous, then f_xy = f_yx.
Partial derivatives appear whenever a quantity depends on more than one input. In thermodynamics, the rate of change of pressure with respect to temperature at constant volume is a partial derivative. In economics, a firm's marginal cost with respect to labour (holding capital fixed) is a partial derivative. Machine-learning gradient descent computes partial derivatives of a loss function with respect to every model parameter.
Students often forget to treat the other variable as a constant and accidentally differentiate both variables at once. If you are finding f_x, every y in the expression is just a number.
Confusing the notation f_yx: the subscript order means differentiate with respect to y first, then x. Read right to left from the function name.
Assuming that "slope in the x-direction" on a contour map means reading along the x-axis. The slope f_x at a point P is about how the function value changes, not about the contour spacing alone. Direction matters.
Forgetting the sign flip when the lower limit of an integral depends on the differentiation variable. If f = ∫ from y to x of g(t) dt, then f_y = −g(y), not +g(y).
⚠️ Expect questions that ask you to find f_x − f_y or similar combinations. Compute each partial separately, then combine.
⚠️ Graphical interpretation questions (contour maps and 3D surfaces) are very commonly tested. Know how to read the sign of f_x and f_y from a picture without any algebra.
⚠️ Mixed partials with the product rule and chain rule together are a favourite. Be methodical: find f_y first, then differentiate that entire expression with respect to x.
⚠️ The fundamental theorem of calculus applied to integrals with variable limits is a reliable exam topic. Watch the sign when differentiating with respect to the lower limit.
True or false: When computing ∂f/∂x, you treat y as a constant. (True.)
If you walk in the y-direction from a point on a contour map and stay on the same contour line, f_y is ______ at that point. (Zero.)
True or false: f_xy and f_yx are always equal. (False, they are equal when the mixed partials are continuous, which is almost always the case in this course.)
Fill in the blank: If f(x,y) = ∫ from y to x of g(t) dt, then f_y = ______. (−g(y).)
True or false: If the surface z = f(x,y) slopes downward as you move in the positive x-direction, then f_x > 0. (False, f_x < 0.)
Q: Given f(x, y) = (x² + 2y)(y² − x), find f_x.
A: Apply the product rule. f_x = 2x(y² − x) + (x² + 2y)(−1) = 2xy² − 2y − 3x².
Q: From a contour map, the contour values decrease as you move in the positive x-direction from P, and P sits on a contour line that runs horizontally. What are the signs of f_x and f_y at P?
A: f_x < 0 (moving in x goes to lower values). f_y = 0 (moving in y keeps you on the same contour).
Q: For f(x, y) = x² cos(xy), find f_yx.
A: First, f_y = −x³ sin(xy). Then differentiate with respect to x: f_yx = −3x² sin(xy) − x³ y cos(xy) = −x²(3 sin(xy) + xy cos(xy)).
Q: Find f_y when f(x, y) = ∫ from y to x of cos(t⁶) dt.
A: Rewrite as −∫ from x to y of cos(t⁶) dt. By the fundamental theorem, f_y = −cos(y⁶).
Q: Find the slope in the x-direction at P(0, 2) for f(x, y) = 5(y² − x²) ln(x + y).
A: f_x = 5[−2x ln(x + y) + (y² − x²)/(x + y)]. At (0, 2): f_x = 5[0 + 4/2] = 10.
Partial derivatives are the building blocks of the gradient vector ∇f = (f_x, f_y), which points in the direction of steepest ascent. That gradient feeds into directional derivatives, tangent planes, and the multivariable chain rule. Later in the course, second-order partials drive the second derivative test for local maxima and minima of multivariable functions.
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