Partial Derivative¶

Formula¶

\[ \frac{\partial f}{\partial x_i}(x)= \lim_{h\to 0}\frac{f(x_1,\dots,x_i+h,\dots,x_n)-f(x)}{h} \]

Parameters¶

\(f:\mathbb{R}^n\to\mathbb{R}\): multivariable function
\(x_i\): \(i\)-th input variable

What it means¶

A partial derivative measures how \(f\) changes with respect to one variable while holding the others fixed.

What it's used for¶

Multivariable optimization.
Constructing gradients and Jacobians.

Key properties¶

One partial derivative per input coordinate.
Can exist even when the function is not fully differentiable.

Common gotchas¶

Holding other variables fixed is essential.
Mixed partials may require smoothness assumptions to be equal.

Example¶

For \(f(x,y)=x^2y+y\), \(\partial f/\partial x = 2xy\) and \(\partial f/\partial y=x^2+1\).

How to Compute (Pseudocode)¶

Input: function f(x1,...,xn), point x, coordinate i, small step h
Output: approximate partial derivative d f / d x_i at x

x_plus <- copy of x
x_minus <- copy of x
x_plus[i] <- x_plus[i] + h
x_minus[i] <- x_minus[i] - h

return (f(x_plus) - f(x_minus)) / (2h)

Complexity¶

Time: \(O(1)\) function evaluations (2 evaluations of \(f\)) for this finite-difference estimate
Space: \(O(n)\) if copies of the input vector are materialized (or \(O(1)\) extra with in-place perturbation/restoration)
Assumptions: Excludes the internal cost of evaluating \(f\); symbolic or automatic differentiation follows different computational workflows