Free lesson · Partial derivatives

Hessians, mixed derivatives and second-order scenarios

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Start with the idea

A straight-line estimate can miss the bend in the function. A second-order estimate also counts each input’s curvature and the interactions between inputs.

Symbols, units & horizon

H: Hessian matrix, not a candle high here
fₓₓ,fᵧᵧ: pure second partial derivatives
fₓᵧ,fᵧₓ: mixed second partials
h=(Δx,Δy): input-change vector
∇f: gradient at the starting point
hᵀHh: quadratic form
Δf: output change
Each Hessian entry: output units divided by the two associated input units

When and why to use this

Use Hessians to inspect local curvature, compare first- and second-order scenario estimates, and understand optimisation conditioning.

Start here · the same idea in everyday words

Technical term	Everyday meaning	Concrete example
Curvature	How a slope changes as its input changes.	A bowl gets steeper as you move away from its centre.
Hessian	A table of second derivatives.	Its diagonal measures each input’s own curvature; off-diagonal entries measure interactions.
Mixed derivative	How the slope for one input changes when another input changes.	A mixed x-y derivative records how the x slope responds to y.

Unpack a technical sentence

Technical wording: Second-order scenarios include curvature and cross-input interactions.

Read it as: A straight-line estimate can miss the bend in the function. A second-order estimate also counts each input’s curvature and the interactions between inputs.

A slower worked example · one step at a time

Use f=x squared plus y squared, with all numbers dimensionless. At x=1 and y=2, the output is 5.
The slopes are 2x=2 and 2y=4. The Hessian has diagonal values 2 and 2, and off-diagonal values zero because the two squared terms have no interaction.
Increase x by .1 and y by .2. The straight-line change is 2 times .1 plus 4 times .2=1, giving a prediction of 6.
The curvature correction is half of [2 times .1 squared plus 2 times .2 squared]=.05. The second-order prediction is 6.05.
Direct evaluation gives 1.1 squared plus 2.2 squared=6.05. The second-order answer is exact here because the function is quadratic. A general curved function can still have higher-order error.

Self-assessed: Does a second-order approximation always give the exact answer?

No. It is exact for this quadratic. More general smooth functions can have third- and higher-order contributions.

For a function with continuous second partials in a neighbourhood, mixed partial derivatives agree.
Without the regularity assumptions that symmetry cannot simply be assumed.
A Hessian is local, just like a gradient.
The second-order multivariable Taylor term is one half of a quadratic form in the input changes.
Off-diagonal terms appear twice in the full sum, so the two-variable cross contribution has no remaining one-half factor when the Hessian is symmetric.

H = (\begin{matrix} f_{x x} & f_{x y} \\ f_{y x} & f_{y y} \end{matrix}), Δ f \approx \nabla f^{𝖳} h + \frac{1}{2} h^{𝖳} H h

Calculus: derivation and arithmetic

Hessians, mixed derivatives and second-order scenarios

For f=x²+xy+2y², compute fₓ=2x+y and fᵧ=x+4y. Differentiate again to obtain H=[[2,1],[1,4]].
Expand hᵀHh=2(Δx)²+2ΔxΔy+4(Δy)². Multiplying by one half gives the curvature correction.
At (1,1), gradient=(3,5). With h=(.1,−.2), the linear term is −.7. Curvature is .01−.02+.08=.07, giving total −.63.

Work it by hand

Original value f(1,1)=4. New value f(1.1,.8)=3.37. The change is exactly −.63 because this function is quadratic and has no higher-order remainder.

Architecture pseudocode

def quadratic_value(x,y): return x*x+x*y+2*y*y

def quadratic_change(x,y,dx,dy):
    linear=(2*x+y)*dx+(x+4*y)*dy
    curvature=dx*dx+dx*dy+2*dy*dy
    return linear,curvature,linear+curvature

print(quadratic_change(1,1,.1,-.2),quadratic_value(1.1,.8)-quadratic_value(1,1))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "start": 1**2+2**2,
        "linear": 5+2*.1+4*.2,
        "correction": .5*(2*.1**2+2*.2**2),
        "actual": 1.1**2+2.2**2,
    }

print(plain_language_example())

An analogy to remember

Bending a sheet in two directions can create an interaction that neither one-direction bend captures. Mixed derivatives quantify that local interaction.

How this becomes a building block

Gamma is one entry of a pricing Hessian. Cross sensitivities such as spot–volatility interaction can matter in a joint market shock. In portfolio optimisation, the covariance quadratic form supplies curvature of a mean–variance objective.

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 quadratic_value(x,y): return x*x+x*y+2*y*y

def quadratic_change(x,y,dx,dy):
    linear=(2*x+y)*dx+(x+4*y)*dy
    curvature=dx*dx+dx*dy+2*dy*dy
    return linear,curvature,linear+curvature

print(quadratic_change(1,1,.1,-.2),quadratic_value(1.1,.8)-quadratic_value(1,1))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "start": 1**2+2**2,
        "linear": 5+2*.1+4*.2,
        "correction": .5*(2*.1**2+2*.2**2),
        "actual": 1.1**2+2.2**2,
    }

print(plain_language_example())

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