Partial Derivative Calculator

A partial derivative measures how a multivariable function changes with respect to one variable while holding the others fixed. For $f(x, y)$:

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

The notation $\partial$ (curly d) distinguishes partial derivatives from ordinary derivatives $\frac{d}{dx}$. Equivalent notations include $f_x$, $\partial_x f$, $D_x f$.

Geometric meaning: $\frac{\partial f}{\partial x}(a, b)$ is the slope of the surface $z = f(x,y)$ at $(a,b)$ in the $x$-direction — the tangent line lies in the plane $y = b$.

Why this matters: gradient descent, optimization, error propagation, and most of vector calculus rest on partial derivatives. The gradient $\nabla f = (f_x, f_y, f_z)$ points in the direction of steepest ascent.

Rule 1: Treat Other Variables as Constants

To find $\frac{\partial f}{\partial x}$, treat $y, z, \ldots$ as constants and differentiate $f$ as a single-variable function of $x$.

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

• $\frac{\partial f}{\partial x} = 2xy$ (the $3y$ vanishes since it has no $x$)
• $\frac{\partial f}{\partial y} = x^2 + 3$ ($x^2$ acts as a coefficient)

Rule 2: Chain Rule and Product Rule Still Apply

For $f(x, y) = \sin(xy)$:

$\frac{\partial f}{\partial x} = \cos(xy) \cdot y$

The $y$ inside the parenthesis is treated as a constant coefficient when differentiating $xy$ with respect to $x$.

Higher-Order Partials

$f_{xx} = \frac{\partial^2 f}{\partial x^2}, \quad f_{xy} = \frac{\partial^2 f}{\partial y \partial x}$

Clairaut's theorem (mixed partials): if $f$ has continuous second partials, then $f_{xy} = f_{yx}$. Order of differentiation doesn't matter.

Gradient and Directional Derivative

The gradient is the vector of all first partials:

$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$

The directional derivative in direction $\mathbf{u}$ (unit vector) is:

$D_\mathbf{u} f = \nabla f \cdot \mathbf{u}$

Maximized when $\mathbf{u}$ points along $\nabla f$ — this is the steepest ascent direction.

Chain Rule (Multivariable)

If $z = f(x, y)$ and $x = x(t), y = y(t)$:

$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$

• Differentiating the wrong variable: Always identify which variable is 'live' and which are held constant. Underlining the live variable in your scratch work helps.
• Forgetting the chain rule: $\frac{\partial}{\partial x}\sin(xy) = y\cos(xy)$, not just $\cos(xy)$.
• Confusing notation: $f_{xy}$ means differentiate first w.r.t. $x$, then $y$ (some books reverse this — check the convention).
• Wrong gradient direction: $\nabla f$ points in the direction of steepest ascent, not motion. To minimize, move opposite to $\nabla f$.
• Mixing partial and total derivatives: When $x$ and $y$ both depend on $t$, use the chain rule — not $\partial f/\partial t$, which is zero if $f$ has no explicit $t$.