Free lesson · Partial derivatives

Gradients, directional derivatives and Jacobians

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Start with the idea

The gradient gathers the partial derivatives of a scalar output into one vector.
Its dot product with an input direction gives the rate in that direction.
A Jacobian does the same bookkeeping for multiple outputs.

Symbols, units & horizon

∇f: gradient of a scalar function
T: transpose
u: direction vector of unit Euclidean length for a per-unit-distance derivative
Dᵤf: directional derivative
Fᵢ: output component i
xⱼ: input component j
J: Jacobian matrix
Δx,ΔF: small finite input and output changes
|| ||: vector length

When and why to use this

Use gradients for scenario directions and optimisation; use Jacobians to connect stages of a multivariable model and audit input/output dimensions.

For a unit direction u, the directional derivative measures output change per unit distance in the chosen coordinate scale.
Rescaling an input changes what a unit step means.
This matters when combining prices, rates and volatility coordinates.
If a model maps an input vector x to an output vector F, its Jacobian entry Jᵢⱼ is the partial derivative of output i with respect to input j.
Multiplying J by a small input change gives a first-order output-change vector.
The rows and columns must match the documented ordering.

\nabla f = (f_{x}, f_{y})^{𝖳}, D_{u} f = \nabla f^{𝖳} u, J_{i j} = \frac{\partial F_{i}}{\partial x_{j}}, Δ F \approx J Δ x

Calculus: derivation and arithmetic

Gradients, directional derivatives and Jacobians

For f=x²+2y², the gradient is (2x,4y). At (1,1) it is (2,4).
Use direction u=(3/5,4/5), whose squared length is 9/25+16/25=1. The directional derivative is 2×3/5+4×4/5=4.4.
For F=(x+y,xy), differentiate each output to get Jacobian rows (1,1) and (y,x). At (2,3), a change (.1,−.2) gives first-order output change (−.1,−.1).

Work it by hand

For F=(x+y,xy), actual change from (2,3) to (2.1,2.8) is (−.1,−.12). The −.02 difference in the second component is the product of the two input changes, omitted at first order.

An analogy to remember

A gradient is a slope map telling you how a hill rises in each coordinate direction. A Jacobian is a table of such responses when you are tracking several outputs at once.

How this becomes a building block

A Jacobian can map market-quote changes into instrument-price changes or map parameter changes into calibration errors. Reverse-mode automatic differentiation efficiently propagates a scalar output’s sensitivities backward through a computational graph; it differentiates the implemented operations, not an independently validated economic truth.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def directional_quadratic(x,y,direction):
    ux,uy=direction
    if abs(ux*ux+uy*uy-1)>1e-9: raise ValueError("Unit direction required")
    return 2*x*ux+4*y*uy

def example_jacobian(x,y):
    return [[1,1],[y,x]]

def linear_output_change(jacobian,change):
    if any(len(row)!=len(change) for row in jacobian): raise ValueError("Dimensions must match")
    return [sum(a*b for a,b in zip(row,change)) for row in jacobian]

print(directional_quadratic(1,1,(.6,.8)),linear_output_change(example_jacobian(2,3),[.1,-.2]))

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