M8-T3 — Partial Derivatives

Learning objectives

After this lesson, you should be able to:

• Compute partial derivatives of functions such as range, bearing and arm kinematics.
• Assemble the gradient and use the first-order (tangent) approximation f(x + δ) ≈ f(x) + ∇f·δ.
• Apply the chain rule for compositions of functions of several variables.
• Check analytic derivatives with finite differences and choose a sensible step size.

Introduction and motivation

A LiDAR measures range r = √(x² + y²) to a landmark. A filter wants to know how the range changes if the robot moves 10 cm east. A solver wants to know how the hand moves if the shoulder joint turns 1°. Both are partial derivatives: the sensitivity of one output to one input with the others frozen. Getting them right (and verifying them numerically) is a daily task, because almost every estimator and controller in robotics is built on linearisation.

The intuition

For a function of one variable the derivative is the slope of the tangent line. For a function of several variables, you can still slice: hold all inputs but one fixed and look at the resulting one-variable function. Its slope is the partial derivative with respect to that input. Several slopes, one per input, define a tangent plane (or hyperplane) that fits the function near the point. Replacing the curved function by that flat model is linearisation.

Geometric and physical interpretation

Geometric view. The gradient ∇f = (∂f/∂x, ∂f/∂y) points in the direction of steepest increase of f, and its length is the slope in that direction. For a vector-valued function each component has its own gradient. Physical view. For the arm, ∂p/∂q₁ is the velocity of the hand per unit angular velocity of joint 1 with joint 2 locked: it is perpendicular to the line from joint 1 to the hand, with length equal to that distance. This is the familiar “velocity = ω × r”.

The mathematics

f(x, y)

a scalar function of two variables (the idea carries to n variables).

∂f/∂x

the partial derivative: differentiate with respect to x, treating y as a constant.

∇f

the gradient, the vector of partial derivatives.

δ

a small change in the inputs.

Definition

∂f/∂x (x, y) = lim_{h→0} [ f(x + h, y) − f(x, y) ] / h

What it means. Nudge only x by h, measure the change in f, divide by h and shrink h. All other inputs stay at their current values.

First-order approximation and the gradient

f(x + δ, y + ε) ≈ f(x, y) + (∂f/∂x) δ + (∂f/∂y) ε = f + ∇f · (δ, ε)

What it means. Near the point, the change in f is a weighted sum of the input changes with the partial derivatives as weights. The error is of second order in the step size.

The chain rule

d/dt f(x(t), y(t)) = (∂f/∂x) ẋ + (∂f/∂y) ẏ        (z = g(f(x)):  ∂z/∂x = (∂g/∂f)(∂f/∂x))

What it means. Derivatives of compositions multiply (and, for vector functions, become matrix products, which is why the Jacobian of a chain is a product of Jacobians).

Partial derivative: slope along one input with the others frozen. Gradient = vector of partials. f(x + δ) ≈ f(x) + ∇f·δ, with error of order δ².

Step-by-step derivation: gradients of range and the arm, and finite differences

1. Range. r = (x² + y²)^(1/2). Using the chain rule: ∂r/∂x = ½(x² + y²)^(−1/2)·2x = x/r and ∂r/∂y = y/r. So ∇r = (x/r, y/r): the unit vector pointing away from the origin. Moving a distance d directly away from the landmark changes the range by exactly d; moving perpendicular changes it (to first order) by zero.
2. Bearing. β = atan2(y, x) gives ∂β/∂x = −y/r² and ∂β/∂y = x/r²: the sensitivity falls as 1/r, so far landmarks give small angular changes per metre.
3. Arm. Differentiate x = l₁ cos q₁ + l₂ cos(q₁ + q₂): ∂x/∂q₁ = −l₁ sin q₁ − l₂ sin(q₁ + q₂), ∂x/∂q₂ = −l₂ sin(q₁ + q₂). Likewise y = l₁ sin q₁ + l₂ sin(q₁ + q₂) gives ∂y/∂q₁ = l₁ cos q₁ + l₂ cos(q₁ + q₂), ∂y/∂q₂ = l₂ cos(q₁ + q₂). The inner angle q₁ + q₂ contributes through the chain rule, which is why q₁ appears in both terms.
4. Finite differences. Replace the limit by a small step. Forward difference: (f(x + h) − f(x))/h has truncation error about (h/2)f″. Central difference: (f(x + h) − f(x − h))/(2h) has error about (h²/6)f‴, much smaller for the same h, at the price of two evaluations.
5. Choosing h. A step that is too large has big truncation error; one that is too small loses digits to rounding (subtracting nearly equal numbers). For central differences in double precision a good h is about ∛ε·(1 + |x|) ≈ 10⁻⁵ · (1 + |x|) (with ε = 2.2 × 10⁻¹⁶).
6. Linearisation error. By Taylor's theorem, f(x + δ) = f(x) + f′(x)δ + ½f″(x)δ² + …, so the error of the tangent approximation is about ½|f″|δ²: halving the step quarters the error.

Gradient checking is a debugging habit. Whenever you write an analytic Jacobian for a filter or optimiser, compare it with central differences at a few random points. Mismatch beyond about 1e-6 relative means an algebra, sign or unit error. This single test catches most bugs in EKF and bundle-adjustment code.

The multivariate chain rule. If y = g(u) and u = f(x), then ∂yᵢ/∂xⱼ = Σₖ (∂yᵢ/∂uₖ)(∂uₖ/∂xⱼ): sum over every path from xⱼ to yᵢ. In matrix form, the Jacobians multiply: J_{g∘f} = J_g J_f. This is how the sensor model Jacobian is built from the Jacobian of the frame transformation and the Jacobian of the polar conversion.

A worked example: gradients, linearisation and a finite-difference check

• A simple function. f(x, y) = x²y: ∂f/∂x = 2xy, ∂f/∂y = x². At (1, 2) the gradient is (4, 1).
• Finite-difference check. A central difference in x at (1, 2) with h = 10⁻⁶: (f(1 + h, 2) − f(1 − h, 2))/(2h) = 4.000000 ✓. A forward difference with h = 0.1 on x² at x = 1 gives (1.21 − 1)/0.1 = 2.1 versus the exact 2: an error of 0.1, equal to h.
• Range gradient. At the landmark (3, 4), r = 5 and ∇r = (3/5, 4/5) = (0.6, 0.8).
• Linearisation. Move the landmark by (0.1, 0): prediction Δr ≈ 0.6·0.1 = 0.06; exact √(3.1² + 4²) − 5 = 0.06063. The error 0.0006 is of order δ² as expected.
• Bearing gradient. ∂β/∂x = −4/25 = −0.16, ∂β/∂y = 3/25 = 0.12 rad per metre.
• Arm partials. For q = (0°, 90°) with l₁ = 1.1, l₂ = 0.8: ∂x/∂q₁ = −0 − 0.8 = −0.8, ∂y/∂q₁ = 1.1 + 0 = 1.1, ∂x/∂q₂ = −0.8, ∂y/∂q₂ = 0. So rotating the shoulder 1° (0.01745 rad) moves the hand by about (−0.0140, 0.0192) m.

Robotics application

SLAM and EKF. The measurement Jacobian of a range-bearing sensor is exactly the matrix of the partial derivatives above, evaluated at the current estimate; see EKF-SLAM and nonlinear least squares.

UAV state estimation. The extended Kalman filter linearises the motion and measurement models with partial derivatives with respect to the state; see the extended Kalman filter.

Manipulators. The columns of the arm Jacobian are the partials of hand position with respect to each joint (next lesson).

LiDAR registration. Gauss-Newton ICP linearises the point-to-plane residual with respect to the pose parameters: the same first-order approximation; see ICP fundamentals.

Engineering implications and limitations

Practical rules

• Verify analytic derivatives numerically. A central-difference check at random states, with relative tolerance about 1e-6, finds most Jacobian bugs.
• Use central differences and a scaled step. h ≈ 1e-5 (1 + |x|) in double precision; forward differences with h ≈ 1e-8 are the fallback when evaluations are expensive.
• Keep units straight. Partial derivatives carry units: metres per radian for the arm, radians per metre for bearing. Mixing them in one matrix needs scaling before norms or condition numbers are meaningful.
• Watch singular points. Range has no derivative at r = 0, atan2 jumps at ±π; guard against dividing by r when r is tiny.
• Linearise only for small steps. The tangent approximation error grows like δ²; re-linearise at every iteration or time step.
• Prefer analytic or automatic differentiation in production. Finite differences are for checking and quick prototypes; they are noisy and cost n extra evaluations.

Limits

The partial derivatives describe the function only near the point. Functions with kinks, such as |x| or contact models, have no derivative there, and strong curvature makes the linear model invalid for all but tiny steps.

Common misconceptions

Engineering takeaways

1. A partial derivative is the slope of f along one input with the others held fixed; the gradient collects them.
2. f(x + δ) ≈ f(x) + ∇f·δ, with error of order δ²: that is linearisation.
3. The chain rule multiplies derivatives; for vector functions it multiplies Jacobians.
4. Range gradient is (x/r, y/r); arm partials come from differentiating the link vectors with the chain rule.
5. Check analytic derivatives with central differences using a scaled step of about 1e-5.

References and further reading

Primary papers and authoritative documentation.

• MIT 18.06SC Linear Algebra (OCW): lecture on linear approximation and partial derivatives (18.02 or 18.06 supplementary)
• Mathematics for Machine Learning: chapter 5, vector calculus: partial differentiation and gradients
• Boyd and Vandenberghe, Introduction to Applied Linear Algebra (VMLS): chapter 8, derivatives and linearisation of vector-valued functions; the site did not load when we last checked
• UCSD ECE276A: Sensing and Estimation in Robotics: gradients and linearisation in estimation and registration

Knowledge check

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What’s next

Stacking the partial derivatives of every output with respect to every input gives the Jacobian matrix: the best local linear approximation of a vector function, and the star of the rest of this module.

Related content

Nothing below is required to finish this lesson.

• Before
  M8-T2: Forward Kinematics of a Two-Link Manipulator

  Compute the hand position of a planar two-link arm and describe its workspace.