Partial Derivatives

Subtopic: Multivariable

Partial derivatives measure how a function of several variables changes when you vary one variable while holding others fixed. They're the building blocks of multivariable calculus—combining to form gradients, Jacobians, and Hessians. Understanding partial derivatives is essential for optimization, physics, and machine learning.

Introduction

A function like ƒ(x,y)=x2+x*y has two inputs. How does ƒ change when x changes? When y changes? These are different questions with different answers. Partial derivatives answer each question separately.

It's like asking how temperature varies with latitude vs longitude—each direction gives a different rate of change.

Definition

The partial derivative of ƒ(x,y) with respect to x is:

∂(ƒ)/∂(x)=(lim_)((ƒ(x+h,y)−ƒ(x,y))/h)

Treat y as a constant and differentiate with respect to x using ordinary derivative rules.

Similarly for ∂(ƒ)/∂(y): treat x as constant, differentiate with respect to y.

Worked Example

Find the partial derivatives of ƒ(x,y)=x3*y+2*x*y2-y.

∂f/∂x (treat y as constant)

∂(ƒ)/∂(x)=3*x2*y+2*y2

The y terms act like constants: (x3*y)′=3*y*x2, (2*x*y2)′=2*y2, (-y)′=0.

∂f/∂y (treat x as constant)

∂(ƒ)/∂(y)=x3+4*x*y−1

Now x is constant: (x3*y)′=x3, (2*x*y2)′=4*x*y, (-y)′=-1.

Geometric Interpretation

For z=ƒ(x,y), the graph is a surface in 3D .

∂(ƒ)/∂(x) at (a,b) is the slope of the slice of the surface where y=b (walking in the x-direction).

∂(ƒ)/∂(y) at (a,b) is the slope of the slice where x=a (walking in the y-direction).

Each partial derivative measures steepness along one coordinate axis.

Higher-Order Partials

You can take partial derivatives of partial derivatives:

(ƒ_xx)=∂2(ƒ)/∂(x2), (ƒ_xy)=∂2(ƒ)/(∂(y)*∂(x)), (ƒ_yy)=∂2(ƒ)/∂(y2)

Mixed partials: (ƒ_xy) means differentiate with respect to x first, then y.

Clairaut's Theorem

If mixed partials are continuous, they're equal:

(ƒ_xy)=(ƒ_yx)

Order doesn't matter (for "nice" functions).

The Gradient

The gradient collects all partial derivatives into a vector:

∇*ƒ=(∂(ƒ)/∂(x),∂(ƒ)/∂(y))

It points in the direction of steepest increase; its magnitude is the steepest slope.

Chain Rule for Partials

If z=ƒ(x,y) and x,y depend on t:

d(z)/d(t)=∂(ƒ)/∂(x)d(x)/d(t)+ ∂(ƒ)/∂(y)d(y)/d(t)

Changes propagate through all paths from t to z.

Applications

In physics, partial derivatives describe how physical quantities depend on multiple variables (pressure, temperature, volume).

In economics, marginal analysis uses partials to measure sensitivity to price, quantity, etc.

In machine learning, gradients (vectors of partials) drive optimization of loss functions.

Summary

Partial derivatives measure the rate of change with respect to one variable, holding others constant. Compute by treating other variables as constants. ∂(ƒ)/∂(x) is the slope along the x-direction. The gradient ∇ƒ collects all partials. Mixed partials are equal ((ƒ_xy)=(ƒ_yx)) when continuous. Chain rules sum contributions from all paths.