Difficulty: Intermediate | Prerequisites: Calculus I and II (derivatives, chain rule, basic integration), familiarity with functions of several variables (Section 14.1).
Sections 14.3 and 14.4 are where single-variable differentiation scales up to functions of two or more variables. You already know how to find the slope of a curve; now you are finding slopes in every direction on a surface, and stitching those slopes together into a flat plane that just touches the surface at one point. This is the gateway to optimisation, error analysis, and modelling in higher dimensions.
If you are comfortable taking ordinary derivatives and using the tangent-line formula y = f(a) + f'(a)(x - a), you have the right foundation. Everything here generalises that idea: hold all variables but one fixed (partial derivatives), then combine the partials to build a tangent plane and a linear approximation (differentials).
Partial derivatives measure how a multivariable function changes when you nudge one input at a time. The tangent plane is the two-variable version of the tangent line: a flat surface that best approximates the graph near a point. Differentials let you use that tangent plane to estimate how small changes in the inputs ripple through to changes in the output, which is exactly how engineers estimate measurement error.
Partial derivative (fx or ∂f/∂x)
The rate of change of f with respect to x while every other variable is held constant. Computed exactly like an ordinary derivative, treating the other variables as numbers.
In simple terms, this means: pick one input knob and turn it slightly; the partial derivative tells you how fast the output moves.
Second-order partial derivative (fxx, fxy, etc.)
The partial derivative of a partial derivative. fxx means differentiate with respect to x twice. fxy means differentiate first with respect to x, then with respect to y.
Think of it as: how the slope itself is changing. fxx is concavity in the x-direction; fxy captures how the x-slope varies as you move in y.
Clairaut's theorem (equality of mixed partials)
If fxy and fyx are both continuous, then fxy = fyx. You can differentiate in either order and get the same result.
In simple terms, this means: for any well-behaved function, the order of mixed partial differentiation does not matter.
Tangent plane
The plane that best approximates the graph z = f(x, y) at a given point. Its equation is z = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b).
Think of it as: the two-variable version of the tangent line. It touches the surface at one point and has the same slope in every direction there.
Differential (dz or df)
The linear part of the change in f: df = fx dx + fy dy (for two variables), or df = fx dx + fy dy + fz dz (for three). It approximates the actual change Δf when dx, dy, dz are small.
In simple terms, this means: a quick, pencil-and-paper estimate of how much the output shifts when the inputs shift by small amounts.
Linearisation (linear approximation)
The function L(x, y) = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b), which approximates f near (a, b). The tangent plane, reframed as an approximation tool.
Relative error
The absolute error divided by the measured (or true) value: |Δx| / |x|. Expressed as a fraction or percentage. Differentials let you propagate relative errors through a formula without plugging in the actual perturbations.
Computing partial derivatives
To find fx, treat y (and any other variables) as constants and differentiate with respect to x using all the usual single-variable rules (product rule, chain rule, etc.).
To find fy, treat x as a constant and differentiate with respect to y.
Example: if f(x, y) = xe^(-y^2) - ye^(-x^2), then fx = e^(-y^2) + 2xye^(-x^2) and fy = -2xye^(-y^2) - e^(-x^2).
Estimating partials from a table
When you have a table of values rather than a formula, approximate the partial derivative with a difference quotient.
Central difference (preferred when data on both sides is available): fx(a, b) ≈ [f(a+h, b) - f(a-h, b)] / 2h.
Forward difference: fx(a, b) ≈ [f(a+h, b) - f(a, b)] / h.
The same logic applies in the y-direction for fy.
Second-order partial derivatives
fxx = ∂/∂x(fx): differentiate the first partial with respect to x again.
fyy = ∂/∂y(fy): differentiate the first partial with respect to y again.
fxy = ∂/∂y(fx): differentiate fx with respect to y. By Clairaut's theorem, fxy = fyx whenever both are continuous.
From a table, approximate fxx by applying the central-difference formula to values of fx (which you already approximated), or use the second-difference formula: fxx(a, b) ≈ [f(a+h, b) - 2f(a, b) + f(a-h, b)] / h^2.
The tangent plane equation
For z = f(x, y) at the point (a, b), the tangent plane is: z = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b).
This is the direct generalisation of the single-variable tangent line y = f(a) + f'(a)(x - a).
Geometric meaning: the tangent plane touches the surface at (a, b, f(a, b)) and has the same slope as the surface in every direction at that point.
Finding tangent planes in practice
Compute fx and fy.
Evaluate f, fx, and fy at the given point (a, b).
Substitute into z = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b).
Simplify if needed.
Parallel tangent planes
Two planes are parallel when their normal vectors are proportional.
The tangent plane to z = f(x, y) at (a, b) has normal vector n =
To find a point on another surface whose tangent plane is parallel to a given plane, set the partial derivatives of the second surface equal to the corresponding coefficients (up to a proportionality constant) and solve the resulting system.
Linear approximation (linearisation)
The linearisation of f at (a, b) is L(x, y) = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b).
For (x, y) close to (a, b), f(x, y) ≈ L(x, y). The approximation improves as (x, y) gets closer to (a, b).
This is the tangent plane repackaged as an approximation tool.
Differentials in two and three variables
For z = f(x, y): dz = fx dx + fy dy.
For w = f(x, y, z): dw = fx dx + fy dy + fz dz.
Here dx, dy, dz are small changes in the input variables; dz (or dw) approximates the resulting change in the output.
Using differentials to approximate a number
Identify a function f and a nearby "nice" point (a, b, c) where f is easy to evaluate.
Set dx = x - a, dy = y - b, dz = z - c.
Compute f(a, b, c) + df as the approximation.
Example: to approximate (1.98)^3 ((3.03)^2 - 1/(1.01)^3), let f(x, y, z) = x^3(y^2 - 1/z^3) and use the base point (2, 3, 1) with dx = -0.02, dy = 0.03, dz = 0.01.
Error propagation with differentials
If each input variable has a maximum absolute error |Δx|, |Δy|, ..., then the maximum absolute error in the output is bounded by |df| ≤ |fx||Δx| + |fy||Δy| + ... (triangle inequality).
For relative error, divide both sides by |f|. A useful shortcut: take the logarithm of the function first, then differentiate. For F = G m1 m2 / r^2, ln F = ln G + ln m1 + ln m2 - 2 ln r, so dF/F = dm1/m1 + dm2/m2 + 2 dr/r. Each variable's relative error adds (with the coefficient being the power it appears with).
In the homework's gravitation problem, each measurement has 3% relative error, and the exponents are 1, 1, 2 respectively, so the maximum relative error in F is 3% + 3% + 2(3%) = 12%.
f_x(a,b) = \lim_{h \to 0} \frac{f(a+h,\, b) - f(a,\, b)}{h}z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) \quad \text{(tangent plane)}dz = f_x\, dx + f_y\, dy \quad \text{(differential, two variables)}dw = f_x\, dx + f_y\, dy + f_z\, dz \quad \text{(differential, three variables)}\text{Central difference: } f_x(a,b) \approx \frac{f(a+h,\, b) - f(a-h,\, b)}{2h}\frac{dF}{F} = \frac{\partial \ln F}{\partial m_1}\, dm_1 + \frac{\partial \ln F}{\partial m_2}\, dm_2 + \frac{\partial \ln F}{\partial r}\, dr \quad \text{(relative error propagation)}Differentials and error propagation are the standard tools for uncertainty analysis in laboratory science and engineering. When a physicist measures masses and distances with known instrument tolerances, the differential formula tells them how confident they can be in the derived force, energy, or field strength. The same method appears whenever a computed quantity depends on several measured inputs: GPS positioning, financial modelling, and medical dosing all use linear error propagation at their core.
Students often think fx means "derivative of f times x." It does not. The subscript names the variable you differentiate with respect to; everything else is held constant.
Students often confuse the tangent plane with the surface itself. The tangent plane is a flat approximation, not the curved surface. It is only accurate near the point of tangency.
When estimating partials from a table, students sometimes use the forward difference when data on both sides of the point is available. The central difference is more accurate and should be your default choice.
Students often forget the factor of 2 on r when propagating relative error through F = Gm1m2/r^2. The exponent on r is -2, so the relative error contribution from r is 2(dr/r), not just dr/r.
⚠️ Tangent plane equations are a near-certainty on exams. Know the formula cold and be able to compute fx and fy quickly for exponential, polynomial, and product-rule functions.
⚠️ Differential approximation problems appear regularly. You must be able to choose a good base point, identify dx/dy/dz, and carry through the algebra.
⚠️ Relative error propagation via logarithmic differentiation is a high-value exam technique. If F = (product of powers), take ln F first, then differentiate.
⚠️ Parallel tangent plane problems test whether you can set up and solve a system of equations from matching normal vectors.
True or False: fx(a, b) measures the rate of change of f in the x-direction while y varies freely.
Answer: False. y is held constant.
Fill in the blank: The tangent plane to z = f(x, y) at (a, b) has normal vector n = < ___, ___, -1 >.
Answer: fx(a, b), fy(a, b).
True or False: If f(x, y, z) = x^2 y z^3, then dF/F = 2 dx/x + dy/y + 3 dz/z.
Answer: True. Take ln f and differentiate.
Fill in the blank: The central-difference approximation of fx(a, b) using step size h is [f(a+h, b) - f(___)] / ___.
Answer: f(a-h, b); 2h.
Q: Find the tangent plane to z = xe^(-y^2) - ye^(-x^2) at the point (2, 1).
A: Compute fx = e^(-y^2) + 2xye^(-x^2), fy = -2xye^(-y^2) - e^(-x^2). At (2, 1): f(2,1) = 2e^(-1) - e^(-4), fx(2,1) = e^(-1) + 4e^(-4), fy(2,1) = -4e^(-1) - e^(-4). Substitute into z = f(2,1) + fx(2,1)(x - 2) + fy(2,1)(y - 1).
Q: Use differentials to approximate (1.98)^3((3.03)^2 - 1/(1.01)^3).
A: Let f(x,y,z) = x^3(y^2 - z^(-3)) with base point (2, 3, 1). Then f(2,3,1) = 8(9 - 1) = 64. Compute fx, fy, fz at that point, set dx = -0.02, dy = 0.03, dz = 0.01, and add df to 64.
Q: If F = Gm1m2/r^2 and each of m1, m2, r has 3% relative error, what is the maximum relative error in F?
A: dF/F = dm1/m1 + dm2/m2 + 2 dr/r = 3% + 3% + 6% = 12%.
Q: Given a table of f(x, y) values at integer grid points, how would you approximate fxy(1, 2)?
A: First approximate fx(1, 1) and fx(1, 3) using central differences in x, then take the central difference of those values in y: fxy(1,2) ≈ [fx(1,3) - fx(1,1)] / 2.
Partial derivatives are the building blocks for every topic that follows in Chapters 14 and 15. The gradient vector (Section 14.6) packages the partials into a single vector that points in the direction of steepest ascent. The chain rule for multivariable functions (Section 14.5) combines partial derivatives with ordinary derivatives when variables depend on other variables. Later, in optimisation (Section 14.7), the tangent plane becomes the starting point for Lagrange multipliers and second-derivative tests.
partial derivative, partial differentiation, tangent plane, linear approximation, linearisation, differential, total differential, error propagation, relative error, absolute error, central difference, forward difference, Clairaut's theorem, mixed partials, second-order partials, fxx, fxy, multivariable calculus, Calculus III, MATH 215, Section 14.3, Section 14.4, functions of several variables, approximation by differentials, Newton's law of gravitation error, uncertainty analysis