Partial Derivative ∂f/∂x

Partial Derivatives

A function of two variables, $f(x, y)$, has two partial derivatives — one for each variable.

$\partial f / \partial x$

The partial derivative with respect to $x$ treats $y$ as a constant and differentiates normally:

$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,\, y) - f(x-h,\, y)}{2h}$

This is the central difference approximation — more accurate than a one-sided difference.

Intuition

• $\partial f / \partial x$ measures the rate of change of $f$ as you move in the $x$-direction
• Think of it as the slope of $f$ along the $x$-axis, holding $y$ frozen

Examples

For $f(x, y) = x^2 + y^2$:

• $\partial f / \partial x = 2x$ (treat $y^2$ as constant → derivative is 0)

For $f(x, y) = x \cdot y$:

• $\partial f / \partial x = y$ (treat $y$ as a constant multiplier)

For $f(x, y) = x^3 + 2xy$:

• $\partial f / \partial x = 3x^2 + 2y$

Your Task

Implement double partial_x(double (*f)(double, double), double x, double y, double h) that approximates $\partial f / \partial x$ at $(x, y)$ using step size $h$.