Partial Derivatives

A partial derivative measures how $f$ changes when one variable moves while the others are held fixed.

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

Notation

$$f_x = \frac{\partial f}{\partial x}, \qquad f_y = \frac{\partial f}{\partial y}, \qquad f_{xy} = \frac{\partial^2 f}{\partial y\,\partial x}$$

Clairaut's theorem — If $f_{xy}$ and $f_{yx}$ are both continuous, then:

f_{xy} = f_{yx}

Geometric interpretation

• $f_x(a,b)$ is the slope of $f$ at $(a,b)$ in the $x$-direction.
• $f_y(a,b)$ is the slope of $f$ at $(a,b)$ in the $y$-direction.

Example — For $f(x,y) = x^{2}y + 3xy^{2}$:

$$f_x = 2xy + 3y^2, \qquad f_y = x^2 + 6xy$$

from sympy import symbols, diff
x, y = symbols('x y')
f = x**2 * y + 3*x * y**2
print("f_x =", diff(f, x))
print("f_y =", diff(f, y))

Key insight: Partial derivatives reduce a multivariable problem to a single-variable one by treating all other variables as constants.

Common pitfall: $\partial f/\partial x$ freezes every other variable — it answers a deliberately narrow question. The real-world change of $f$ when several inputs move at once is the chain rule’s job, not a single partial’s.