Partial Derivatives

Definition

For $f(x, y)$, the partial derivative with respect to $x$ is:

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

Hold $y$ constant and differentiate with respect to $x$.

Alternative notation: $f_x$, $\partial_x f$.

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Higher Derivatives

Second partial derivatives:

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

Mixed partials:

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

Equality of mixed partials: For well-behaved functions (continuous second derivatives):

$f_{xy} = f_{yx}$

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Chain Rule

One Independent Variable

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

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

Two Independent Variables

If $f = f(x, y)$ and $x = x(u, v)$, $y = y(u, v)$:

$\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial u}$

$\frac{\partial f}{\partial v} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial v}$

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Implicit Differentiation

For $F(x, y) = 0$:

$\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}$

Proof: Differentiate $F(x, y(x)) = 0$:

$\frac{\partial F}{\partial x} + \frac{\partial F}{\partial y}\frac{dy}{dx} = 0$

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Total Differential

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

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

This is the total differential: the linear approximation to the change in $f$.

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Directional Derivative

The rate of change of $f$ in direction $\hat{\mathbf{n}} = (n_x, n_y)$:

$D_{\hat{\mathbf{n}}}f = \nabla f \cdot \hat{\mathbf{n}} = \frac{\partial f}{\partial x}n_x + \frac{\partial f}{\partial y}n_y$

The gradient $\nabla f$ points in direction of steepest increase.

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Taylor Series for Two Variables

$f(a+h, b+k) = f(a,b) + hf_x(a,b) + kf_y(a,b) + \frac{1}{2}(h^2f_{xx} + 2hkf_{xy} + k^2f_{yy}) + \cdots$

Compact form:

$f(\mathbf{a} + \mathbf{h}) = f(\mathbf{a}) + \nabla f \cdot \mathbf{h} + \frac{1}{2}\mathbf{h}^T H \mathbf{h} + \cdots$

where $H$ is the Hessian matrix:

$H = \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix}$

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Reciprocity and Cyclic Relations

Reciprocity Relation

For $f(x, y, z) = 0$:

$\left(\frac{\partial x}{\partial y}\right)_z = \frac{1}{\left(\frac{\partial y}{\partial x}\right)_z}$

The subscript indicates what variable is held constant.

Cyclic Relation

$\left(\frac{\partial x}{\partial y}\right)_z \cdot \left(\frac{\partial y}{\partial z}\right)_x \cdot \left(\frac{\partial z}{\partial x}\right)_y = -1$

Application: Thermodynamics (relating partial derivatives of state functions).

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Why Partial Derivatives Matter

• Optimisation: Find maxima and minima of multivariable functions
• Thermodynamics: Relate state variables
• PDEs: Model heat, waves, diffusion
• Machine learning: Gradient descent optimises loss functions

The key skill is determining which variables are independent vs dependent — this determines which derivatives make sense.