Difficulty: Intermediate
Prerequisites: Single-variable derivatives and the chain rule. Familiarity with functions of two variables and their graphs (Section 14.1).
Partial derivatives are the multivariable version of the ordinary derivative. Instead of asking "how does f change as x changes?" for a function of one variable, you now hold one variable fixed and differentiate with respect to the other. This gives you two rates of change at every point. From there, the tangent plane generalises the tangent line: it is the flat surface that best approximates f(x, y) near a point. Linear approximation uses that tangent plane to estimate function values nearby, which is faster than computing the exact value and often good enough for practical purposes.
Partial derivatives measure how a function of two variables changes when you move in one coordinate direction while keeping the other fixed. The tangent plane at a point uses both partial derivatives to give the best flat approximation to the surface. Linear approximation plugs a nearby point into the tangent-plane equation instead of the original function.
Partial derivative with respect to x, f_x or delf/delx
The derivative of f(x, y) with respect to x, treating y as a constant. Think of it as the slope of the surface in the x-direction at a given point.
Partial derivative with respect to y, f_y or delf/dely
The derivative of f(x, y) with respect to y, treating x as a constant. The slope of the surface in the y-direction.
Tangent plane
The plane that best approximates the surface z = f(x, y) near a point (a, b, f(a, b)). It is the 2D analogue of the tangent line. Its equation is z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b).
Linear approximation (linearisation)
Using the tangent plane to estimate f(x, y) near the point of tangency. In simple terms, you replace the curved surface with its tangent plane, which is much easier to evaluate.
Differentiable (at a point)
A function f(x, y) is differentiable at (a, b) if the tangent plane actually gives a good approximation, meaning the error goes to zero faster than the distance to (a, b). If both partial derivatives exist and are continuous near (a, b), the function is differentiable there.
Laplace's equation
The PDE u_xx + u_yy = 0. A function satisfying it is called harmonic. In simple terms, a harmonic function is one whose average value on any small circle equals its value at the centre.
Harmonic function
A twice-differentiable function satisfying Laplace's equation. These appear throughout physics: steady-state heat distribution, electrostatics, fluid flow.
To find f_x: treat y as a constant, differentiate with respect to x using all the usual single-variable rules (product rule, chain rule, etc.).
To find f_y: treat x as a constant, differentiate with respect to y.
Higher-order partials: f_xx means differentiate f_x with respect to x again. f_xy means differentiate f_x with respect to y. Clairaut's theorem says f_xy = f_yx when both mixed partials are continuous.
The homework gives a table of wave heights h = f(v, t) depending on wind speed v (knots) and duration t (hours).
delf/delv measures how wave height changes as wind speed increases, with duration held fixed. Units: feet per knot.
delf/delt measures how wave height changes as the wind blows longer, with speed held fixed. Units: feet per hour.
To estimate f_v(40, 15) from the table, use the values at v = 30 and v = 50 with t = 15: f_v(40, 15) is approximately (f(50, 15) - f(30, 15)) / (50 - 30) = (36 - 16) / 20 = 1.0 feet per knot.
To estimate f_t(40, 15), use t = 10 and t = 20 with v = 40: f_t(40, 15) is approximately (f(40, 20) - f(40, 10)) / (20 - 10) = (28 - 21) / 10 = 0.7 feet per hour.
The tangent plane to z = f(x, y) at the point (a, b) is: z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b).
Linear approximation: to estimate f(x, y) near (a, b), plug (x, y) into the tangent-plane equation instead of the original function.
Let f(x, y) = x^3 + y^3 - 6xy. The point to linearise around is (a, b) = (1, 2).
f(1, 2) = 1 + 8 - 12 = -3.
f_x = 3x^2 - 6y, so f_x(1, 2) = 3 - 12 = -9.
f_y = 3y^2 - 6x, so f_y(1, 2) = 12 - 6 = 6.
Linear approximation: f(0.99, 2.01) is approximately -3 + (-9)(-0.01) + (6)(0.01) = -3 + 0.09 + 0.06 = -2.85.
f(1, 2) = 1 * e^4 - 2 * e^1 = e^4 - 2e.
f_x = e^(y^2) - 2xye^(x^2), so f_x(1, 2) = e^4 - 4e.
f_y = 2xye^(y^2) - e^(x^2), so f_y(1, 2) = 4e^4 - e.
Tangent plane: z = (e^4 - 2e) + (e^4 - 4e)(x - 1) + (4e^4 - e)(y - 2).
For z = x^2 - y^2, the tangent plane at (a, b) has normal vector (-2a, 2b, 1). For this to be parallel to the plane found above, the normal vectors must be proportional. Match the coefficients of (x - a) and (y - b) with those from the previous tangent plane to find a and b.
To check whether u(x, y) satisfies u_xx + u_yy = 0, compute both second partial derivatives and add them.
u = x^2 - y^2: u_xx = 2, u_yy = -2, sum = 0. Harmonic.
u = e^(-x) cos y - e^(-y) cos x: compute each term carefully with the product/chain rule. Check whether the sum is zero.
u = sin(kx) sin(aky): u_xx = -k^2 sin(kx) sin(aky), u_yy = -a^2 k^2 sin(kx) sin(aky), sum = -k^2(1 + a^2) sin(kx) sin(aky). This is zero only if the sine is identically zero, so generally not harmonic.
u = ln(sqrt(x^2 + y^2)): this equals (1/2) ln(x^2 + y^2). Computing the partials gives u_xx + u_yy = 0 for (x, y) != (0, 0). Harmonic away from the origin.
f_x(a, b) = \lim_{h \to 0} \frac{f(a+h, b) - f(a, b)}{h}f_y(a, b) = \lim_{h \to 0} \frac{f(a, b+h) - f(a, b)}{h}\text{Tangent plane: } z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)\text{Linear approximation: } f(x,y) \approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)\text{Laplace's equation: } \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0Numerical estimation of partial derivatives from a table: use the central difference formula where possible, (f(a + h) - f(a - h)) / (2h), or a one-sided difference if data is only available on one side.
Students often think that f_x means "the derivative of f." It does not: it is the derivative with respect to x only, with y held constant. The function still depends on both variables.
Students sometimes apply the product rule between x and y when computing f_x, treating y as a variable. When finding f_x, the variable y is a constant, so terms like 3y^2 are just coefficients.
Students sometimes believe that if both partial derivatives exist at a point, the function must be continuous there. This is false: a function can have partial derivatives at (0, 0) without being continuous at (0, 0). Problem 8 on the homework illustrates exactly this.
Students sometimes confuse the tangent plane with the level surface. The tangent plane is a flat surface touching z = f(x, y) at one point. A level surface is defined by f(x, y, z) = k.
Computing partial derivatives from formulas is bread-and-butter exam material. Expect functions with exponentials, trig, and products.
Estimating partial derivatives from a data table (like the wave-height table) is a classic exam question. Know both the central-difference and one-sided-difference methods.
Writing the equation of a tangent plane at a given point is tested frequently. The formula has three parts: the function value, the f_x term, and the f_y term.
Linear approximation questions typically give you a point near nice integer values and ask for an estimate.
Laplace's equation appears in later courses (PDEs, physics, engineering). Verifying whether a function is harmonic is a straightforward computation that exams like to test.
True or false: To compute f_x for f(x, y) = x^2 y + y^3, you treat y as a constant and get 2xy.
True.
Fill in the blank: The tangent plane to z = f(x, y) at (a, b) is z = f(a, b) + f_x(a, b)() + f_y(a, b)().
(x - a) and (y - b).
True or false: If f_xy and f_yx are both continuous, Clairaut's theorem guarantees they are equal.
True.
True or false: u(x, y) = x^2 + y^2 satisfies Laplace's equation.
False. u_xx + u_yy = 2 + 2 = 4, not zero.
Fill in the blank: To estimate a partial derivative from a table, the simplest method is the ______ formula.
Difference quotient (or central difference).
Q: Find the tangent plane to z = x^2 + xy - y^2 at the point (1, 1, 1).
A: f_x = 2x + y, so f_x(1, 1) = 3. f_y = x - 2y, so f_y(1, 1) = -1. Tangent plane: z = 1 + 3(x - 1) - 1(y - 1), i.e. z = 3x - y - 1.
Q: Use linear approximation to estimate f(2.02, 0.97) given f(2, 1) = 5, f_x(2, 1) = 3, f_y(2, 1) = -2.
A: f(2.02, 0.97) is approximately 5 + 3(0.02) + (-2)(-0.03) = 5 + 0.06 + 0.06 = 5.12.
Q: Verify that u(x, y) = x^2 - y^2 is harmonic.
A: u_x = 2x, u_xx = 2. u_y = -2y, u_yy = -2. u_xx + u_yy = 2 + (-2) = 0. Yes, it satisfies Laplace's equation.
Q: The wave height table gives f(30, 15) = 16 and f(50, 15) = 36. Estimate f_v(40, 15) and interpret.
A: f_v(40, 15) is approximately (36 - 16) / (50 - 30) = 1.0 feet per knot. When t = 15 hours and v = 40 knots, increasing the wind speed by one knot raises the wave height by about 1 foot.
Partial derivatives are the building blocks for the gradient vector (Section 14.6), the chain rule for multivariable functions (Section 14.5), directional derivatives, and optimization (Sections 14.7-14.8). Tangent planes generalise to linear approximation in higher dimensions and underpin the total differential. Laplace's equation reappears in courses on PDEs, electromagnetism, and fluid dynamics.
Partial derivative, partial differentiation, tangent plane equation, linearisation, linear approximation, differentiability, Clairaut's theorem, mixed partial derivatives, second-order partial derivatives, Laplace equation, harmonic function, Stewart Calculus 14.3 14.4, MATH 215 UMich, Calc III tangent planes