Partial Derivative ∂f/∂y

Partial Derivative with Respect to y

Just as $\partial f / \partial x$ differentiates with $x$ varying and $y$ fixed, $\partial f / \partial y$ differentiates with $y$ varying and $x$ fixed:

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

Symmetry Between Variables

For a "symmetric" function like $f(x,y) = x^2 + y^2$, we get:

• $\partial f / \partial x = 2x$
• $\partial f / \partial y = 2y$

They have the same form — $x$ and $y$ play identical roles.

Mixed Partials

Higher-order mixed partials $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$ for smooth functions (Schwarz's theorem). This symmetry is a powerful fact used throughout multivariable calculus.

Examples

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

• $\partial f / \partial y = x^2 + \cos(y)$

For $f(x, y) = e^x \cdot y^2$:

• $\partial f / \partial y = 2y \cdot e^x$

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

• $\partial f / \partial y = 6xy$

Your Task

Implement double partial_y(double (*f)(double, double), double x, double y, double h) that approximates $\partial f / \partial y$ at $(x, y)$ using central differences.