Partial Derivatives

Definition

For a function $f(x,y)$ of several variables, the partial derivative with respect to one variable is computed by differentiating with respect to that variable while holding all others fixed.

Intuition

A partial derivative measures the slope of the function’s graph along a slice parallel to one coordinate axis — it is the ordinary derivative in one direction, treating the other variables as constants.

Formal Description

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

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

$$\frac{\partial f}{\partial x}=6x^2{y}^2,\frac{\partial f}{\partial y}=4x^{3}y+3y^2.$$

Second partial derivatives:

$$\frac{{\partial }^{2}f}{\partial x^2}=12x{y}^2,\frac{{\partial }^{2}f}{\partial y^2}=4x^3+6y.$$

Symmetry of mixed partials (Clairaut’s theorem, when partials are continuous):

$$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}.$$

Total differential:

$$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy.$$

Multivariable chain rule — if $f=f(x(t),y(t))$:

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

If $f=f(x(r,\theta ),y(r,\theta ))$:

$$\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r},\frac{\partial f}{\partial \theta }=\frac{\partial f}{\partial x}\frac{\partial x}{\partial \theta }+\frac{\partial f}{\partial y}\frac{\partial y}{\partial \theta }.$$

Applications

• Gradient vector $\nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)$ used in optimisation and gradient descent.
• Hessian matrix of second partials used in second-order optimisation methods.
• PDEs (heat equation, wave equation, etc.) are expressed in terms of partial derivatives.

Trade-offs

• Existence of all partial derivatives at a point does not imply differentiability or even continuity there.
• Clairaut’s theorem (symmetry of mixed partials) requires continuity of the second partials; pathological counterexamples exist without this condition.

Links

• Chain Rule
• Definition of the Derivative