Partial Derivatives
“A partial derivative is 'hold everything else fixed — how steep is just this one direction?'”
The formula
z = f(x, y) → ∂f/∂x, ∂f/∂yHow to read it: In a function of several variables, the instantaneous slope when only x is nudged while every other variable is held fixed
- ∂
- — The partial sign — 'differentiate in this one direction only'
- ∂f/∂x
- — Partial derivative with respect to x — slope treating y as a constant
- y
- — The other variable, held constant this time
The hook
You stand on a hillside. 'How steep is it if I walk east?' and 'How steep if I walk north?' are different numbers — a partial derivative is the slope in just one direction.
In plain words
For a function of two or more variables, you freeze every variable except the one you care about, then take an ordinary derivative.
The intuition
Picture a bumpy terrain f(x,y). ∂f/∂x is the slope if you step only east-west; ∂f/∂y is the slope if you step only north-south. Even at a single point the steepness depends on direction, so you measure each direction separately.
How it's built
The mechanics are identical to ordinary differentiation, with one extra rule: every variable except the one you are differentiating is treated as a plain number (a constant). So when doing ∂/∂x, you handle y like it were the number 3.
Example
Let f(x,y)=x²y. Differentiating in x (y held constant): ∂f/∂x = 2xy. Differentiating in y (x² held constant): ∂f/∂y = x². At the point (2,3): ∂f/∂x = 2·2·3 = 12 and ∂f/∂y = 2² = 4.
Common mistake
It is tempting to think y also moves while you compute ∂f/∂x. It does not — in a partial derivative y is completely frozen. The case where y moves along with x is handled by a different tool, the total derivative.
Where it's used
Gradient descent in machine learning (a partial derivative for each of thousands of weights = the gradient), partial differential equations for heat and fluids, marginal utility in economics, field calculations in physics — any world with several variables runs on partials.
Where it came from
In the 1700s Euler and d'Alembert refined the idea of differentiating one variable at a time while studying vibrating strings and heat flow. The ∂ symbol, a 'rounded d', was introduced to distinguish it from the ordinary d.
Prerequisites
Quick check
What is the partial derivative of f(x,y)=x²y with respect to y?
- 2xy
- x²✓
- 2x
- y
Practice
For f(x,y)=x²+y², find ∂f/∂x at the point (3,5).
Answer: 6
- Treat y as constant, differentiate in x: ∂f/∂x = 2x
- Plug in x=3 → 2·3 = 6
Takeaway: A partial derivative freezes all but one variable — then it is just ordinary differentiation.
For f(x,y)=x²y, find ∂f/∂x at the point (2,3).
Answer: 12
- Treat y as constant, differentiate in x: ∂f/∂x = 2xy
- Plug in (2,3) → 2·2·3 = 12
Takeaway: When differentiating in x, y stays along for the ride as a number.
For f(x,y)=3x+4y, find ∂f/∂y.
Answer: 4
- Treat x as constant, so 3x differentiates to 0
- 4y differentiated in y → 4
Takeaway: Terms without the target variable are constants and vanish.
Explain in your notebook why it is valid to treat y as a constant when computing ∂f/∂x.
Answer: undefined
- Definition: in f(x+h,y)−f(x,y), y is held identical in both terms
- Only x moves by h → y never changes, so it acts as a constant
- Geometrically you slice an 'east-west cross-section' and measure that slice's slope
Takeaway: The definition itself — slope of a one-direction slice — is exactly what 'hold the other variables fixed' means.