Free lesson · Partial derivatives

Partial derivatives: holding the other inputs fixed

Open interactive lesson Practice calculations Explore labs

Start with the idea

A multivariable function depends on more than one input.
A partial derivative measures a change along one coordinate while the others remain fixed.
It answers a deliberately narrow question about the model.

Symbols, units & horizon

f(x,y): output depending on two inputs
∂: partial differentiation, holding other coordinates fixed
fₓ,fᵧ: partial derivatives
t: path parameter, often time
dx/dt,dy/dt: coordinate rates along that path
df/dt: total output rate along the path
Partial units: output units divided by the chosen input units
V: instrument price
S: spot price
σ(S): volatility specified as a function of spot
V_S: spot partial at fixed volatility
V_σ: price change per unit volatility
σ′(S): volatility change per unit spot

When and why to use this

Use partial derivatives for controlled “change one input” questions, and total derivatives when the inputs are linked along a specified path.

For f(x,y)=x²y+3y², freeze y while differentiating with respect to x, or freeze x while differentiating with respect to y.
“Holding fixed” is part of the definition, not a claim that the real-world variables can never move together.
A total change is different.
If both coordinates depend on time, each changing input contributes through its own partial derivative.
The multivariable chain rule combines those contributions.
The same distinction appears between an option’s fixed-volatility spot sensitivity and its response when implied volatility changes with spot.

f_{x} = \frac{\partial f}{\partial x} = 2 x y, f_{y} = \frac{\partial f}{\partial y} = x^{2} + 6 y, \frac{d f}{d t} = f_{x} \frac{d x}{d t} + f_{y} \frac{d y}{d t}

Calculus: derivation and arithmetic

Partial derivatives: holding the other inputs fixed

Treat y as a constant: derivative of x²y is 2xy, and derivative with respect to x of 3y² is zero.
Treat x as a constant: derivative with respect to y of x²y is x², while derivative of 3y² is 6y.
For a small time step, df≈fₓ dx+fᵧ dy. Divide by dt and take the differentiable path limit to obtain the total derivative.

Work it by hand

At x=2,y=1, fₓ=4 and fᵧ=10. Along a path with dx/dt=3 and dy/dt=−1, total rate is 4×3+10×(−1)=2 output units per time unit.

An analogy to remember

A room’s temperature depends on both location and time. Moving east at one instant measures one partial derivative; walking while the room warms involves both changes.

How this becomes a building block

Partials are individual risk sensitivities. A full scenario uses several of them together. If price V depends on spot S and volatility σ(S), total spot sensitivity includes V_S+V_σ σ′(S). This is a model chain rule, not evidence that a particular volatility–spot relationship is stable.

Further reading: OpenStax Calculus, Volume 3 — partial derivatives ↗

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 example_partials(x,y):
    return 2*x*y,x*x+6*y

def total_rate(x,y,dx_dt,dy_dt):
    fx,fy=example_partials(x,y)
    return fx*dx_dt+fy*dy_dt

def linked_spot_sensitivity(fixed_vol_delta,vega,dvol_dspot):
    return fixed_vol_delta+vega*dvol_dspot

print(example_partials(2,1),total_rate(2,1,3,-1))

Continue learning

Multivariable Calculus & Partial Derivatives — all lessons

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations