Source: Stewart's Multivariable Calculus, §14.4–14.5 | Homework Set 5
Difficulty: Intermediate | Prerequisites: Partial derivatives (§14.3), single-variable chain rule, basic differentiation rules
Sections 14.4 and 14.5 extend differentiation into multiple dimensions. The chain rule tells you how to differentiate composite functions when several variables depend on other variables, which is the engine behind every related-rates problem in multivariable calculus. Implicit differentiation lets you find partial derivatives of a variable defined implicitly by an equation, without solving for it first. Differentials give you a linear approximation of how a function's output changes when its inputs shift by small amounts, which is the basis for error estimation. You should already be comfortable with partial derivatives and the single-variable chain rule before diving in.
The multivariable chain rule lets you find how a quantity changes when it depends on variables that themselves change, which is how you solve related-rates problems in higher dimensions. Implicit differentiation finds partial derivatives straight from an equation F(x,y,z) = 0, no rearranging required. Differentials approximate how small measurement errors propagate through a formula, giving you maximum error estimates.
Multivariable chain rule
If z = f(x, y) and x and y are each functions of one or more other variables (say t, or s and t), then dz/dt is computed by summing the contributions from each path through the dependency tree: dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt).
In simple terms, this means: when z depends on x and y, and x and y each depend on t, you add up how z changes through x and how z changes through y to get the total rate of change.
Implicit differentiation (multivariable)
Given an equation F(x, y, z) = 0 that implicitly defines z as a function of x and y, the partial derivatives are ∂z/∂x = -(Fx / Fz) and ∂z/∂y = -(Fy / Fz), provided Fz ≠ 0.
Think of it as: instead of solving for z and then differentiating, you differentiate the whole equation and solve for the partial derivative algebraically.
Total differential
For z = f(x, y), the total differential is dz = (∂f/∂x) dx + (∂f/∂y) dy. This is the linear approximation of how z changes when x and y each shift by small amounts dx and dy.
In simple terms, this means: the total change in z is roughly the sum of each partial derivative times the corresponding small change in input.
Relative error
The relative error in a quantity Q is |dQ/Q|, expressed as a fraction or percentage. For a product/quotient formula, relative errors in the inputs combine in a predictable way via logarithmic differentiation.
Think of it as: percentage error in the output, estimated from the percentage errors in each measurement.
If z = f(x, y), where x = g(t) and y = h(t), then:
dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)
If z = f(x, y), where x = g(s, t) and y = h(s, t), then:
∂z/∂s = (∂f/∂x)(∂x/∂s) + (∂f/∂y)(∂y/∂s)
∂z/∂t = (∂f/∂x)(∂x/∂t) + (∂f/∂y)(∂y/∂t)
The pattern generalises: for each independent variable, sum over all intermediate variables, multiplying the partial of f with respect to each intermediate variable by the partial (or ordinary derivative) of that intermediate variable with respect to the independent one.
Draw the dependency tree: put f at the top, the intermediate variables (x, y) in the middle, and the independent variables (s, t) at the bottom.
Each path from f down to an independent variable contributes one product of derivatives.
The total derivative with respect to an independent variable is the sum over all such paths.
Identify the quantity whose rate of change you need.
Write an equation relating all the relevant variables.
Differentiate both sides with respect to time (or the relevant independent variable) using the chain rule.
Substitute known values and rates, then solve for the unknown rate.
The area of a triangle with sides a, b and included angle θ is A = (1/2) ab sin θ.
Given: da/dt = 3 cm/s, db/dt = -2 cm/s, dA/dt = 0 (area constant), a = 20 cm, b = 30 cm, θ = π/6.
Differentiate with respect to t:
0 = (1/2)[(da/dt) b sin θ + a (db/dt) sin θ + ab cos θ (dθ/dt)]
Substitute:
0 = (1/2)[(3)(30) sin(π/6) + (20)(-2) sin(π/6) + (20)(30) cos(π/6)(dθ/dt)]
0 = (1/2)[90(1/2) + (-40)(1/2) + 600(√3/2)(dθ/dt)]
0 = (1/2)[45 - 20 + 300√3 (dθ/dt)]
0 = 25/2 + 150√3 (dθ/dt)
dθ/dt = -25 / (300√3) = -1/(12√3) = -√3/36 rad/s
The angle is decreasing at √3/36 radians per second (about 0.048 rad/s).
Given F(x, y, z) = 0 with Fz ≠ 0:
∂z/∂x = -(Fx / Fz)
∂z/∂y = -(Fy / Fz)
This comes from differentiating F(x, y, z) = 0 with respect to x (treating y as constant), using the chain rule on z(x, y):
Fx + Fz (∂z/∂x) = 0, so ∂z/∂x = -Fx / Fz
The equation defines z implicitly (you cannot easily solve for z in closed form).
You need ∂z/∂x or ∂z/∂y at a specific point.
Given: xz + y ln(x - 1) + e^z = 1, find z, ∂z/∂x, ∂z/∂y at (x, y) = (2, 1).
Step 1: find z. Substitute x = 2, y = 1:
2z + (1) ln(2 - 1) + e^z = 1
2z + 0 + e^z = 1
Try z = 0: 0 + e^0 = 1. Yes. So z = 0.
Step 2: set up F. Let F(x, y, z) = xz + y ln(x - 1) + e^z - 1.
Fx = z + y/(x - 1)
Fy = ln(x - 1)
Fz = x + e^z
Step 3: evaluate at (2, 1, 0).
Fx = 0 + 1/(2 - 1) = 1
Fy = ln(1) = 0
Fz = 2 + e^0 = 3
Step 4: apply the formulas.
∂z/∂x = -Fx / Fz = -1/3
∂z/∂y = -Fy / Fz = 0/3 = 0
For z = f(x, y): dz = fx dx + fy dy
For a function of n variables: df = (∂f/∂x1) dx1 + (∂f/∂x2) dx2 + ... + (∂f/∂xn) dxn
Each dx represents a small change (or error) in the corresponding variable.
For functions built from products, quotients, and powers, take the natural log first, then differentiate. This converts multiplicative structure into additive structure.
For F = G m1 m2 / r², take ln:
ln F = ln G + ln m1 + ln m2 - 2 ln r
Differentiate:
dF/F = dm1/m1 + dm2/m2 - 2 dr/r
The maximum relative error uses absolute values:
|dF/F| ≤ |dm1/m1| + |dm2/m2| + 2|dr/r|
Given: m1 = 7 × 10²⁴ kg (0.2% error), m2 = 3 × 10²³ kg (0.3% error), r = 2 × 10⁹ m (0.7% error).
Maximum relative error in F:
|dF/F| ≤ 0.2% + 0.3% + 2(0.7%) = 0.2% + 0.3% + 1.4% = 1.9%
The factor of 2 in front of the r term comes from the exponent in r². This is the key insight: variables raised to higher powers contribute proportionally more to the error.
The maximum relative error in the computed force is 1.9%.
\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}, \quad \frac{\partial z}{\partial y} = -\frac{F_y}{F_z}dz = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy\left|\frac{dF}{F}\right| \leq \left|\frac{dm_1}{m_1}\right| + \left|\frac{dm_2}{m_2}\right| + 2\left|\frac{dr}{r}\right|Engineers use the chain rule constantly when quantities depend on multiple changing inputs, for instance computing how fuel consumption changes with altitude and speed simultaneously. Differentials and error propagation are the backbone of experimental science: every measured quantity has uncertainty, and differentials tell you how those uncertainties combine in your final calculated result.
Students often forget the chain rule entirely when differentiating implicitly, writing ∂F/∂x = 0 instead of Fx + Fz(∂z/∂x) = 0. The z depends on x, so it picks up a ∂z/∂x factor.
When computing relative error, students frequently forget to multiply the relative error in r by the exponent. If F depends on r², the relative error contribution from r is 2|dr/r|, not |dr/r|.
In related-rates problems, students sometimes differentiate before writing down the connecting equation. Always start with the geometric or physical relationship, then differentiate.
Students mix up which variables are held constant in implicit differentiation. When finding ∂z/∂x, you treat y as constant, not the other way round.
⚠️ Related-rates problems almost always appear on exams. The triangle area formula A = (1/2)ab sin θ is a favourite because it is not a right triangle.
⚠️ Implicit differentiation with the formula ∂z/∂x = -Fx/Fz is a common exam shortcut. Know it cold.
⚠️ Error estimation via differentials is frequently tested as a standalone problem. Remember: take ln first for products and quotients, then the relative errors add.
⚠️ Tree diagrams for the chain rule are not just a study aid; some exams ask you to draw them explicitly.
True or False: If z = f(x, y) and x = g(s, t), y = h(s, t), then ∂z/∂s = (∂f/∂x)(∂x/∂s). False – you must also add (∂f/∂y)(∂y/∂s).
Fill in the blank: Given F(x, y, z) = 0, ∂z/∂x = ____ . Answer: -Fx/Fz
True or False: The relative error in r³ is 3 times the relative error in r. True.
Fill in the blank: The total differential of f(x, y) is df = ____ . Answer: fx dx + fy dy
True or False: In a related-rates problem, you should substitute known values before differentiating. False – differentiate first, then substitute.
Q: Two sides of a triangle have lengths a = 10 and b = 15 with included angle θ = π/4. If a increases at 2 cm/s, b decreases at 1 cm/s, and the area is constant, find dθ/dt.
A: Differentiate A = (1/2)ab sin θ with dA/dt = 0. After substituting: dθ/dt = -[b(da/dt) + a(db/dt)] sin θ / (ab cos θ) = -15(2) + 10(-1) / (150 · √2/2) = -(20)(√2/2) / (75√2) = -20/150 = -2/15 rad/s.
Q: Given x²y + yz - xz² = 3, find ∂z/∂x at the point (1, 2, 1).
A: Set F = x²y + yz - xz² - 3. Then Fx = 2xy - z², Fz = y - 2xz. At (1, 2, 1): Fx = 2(1)(2) - 1 = 3, Fz = 2 - 2(1)(1) = 0. Since Fz = 0, the implicit function theorem does not apply at this point (∂z/∂x is undefined here).
Q: The volume of a cylinder is V = πr²h. If r is measured as 5 cm with a 1% error and h as 12 cm with a 2% error, estimate the maximum relative error in V.
A: ln V = ln π + 2 ln r + ln h, so |dV/V| ≤ 2|dr/r| + |dh/h| = 2(1%) + 2% = 4%.
The chain rule is the foundation for directional derivatives (§14.6), which measure rates of change in arbitrary directions. Implicit differentiation connects directly to the gradient vector, since ∂z/∂x = -Fx/Fz is really a ratio of gradient components. Differentials reappear in linear approximation and tangent plane problems (§14.4), and the error-estimation technique shows up again in physics and engineering courses wherever measurements have uncertainty.
Multivariable chain rule, chain rule for partial derivatives, tree diagram derivatives, related rates multivariable, implicit differentiation partial derivatives, implicit function theorem, Fx/Fz formula, total differential, linear approximation, error propagation, relative error, percentage error, logarithmic differentiation for error, differential estimates, MATH 215, Stewart §14.4, Stewart §14.5, Calculus III