Functions of Many Variables
In the real world, things depend on more than one factor. The pressure of a gas depends on both its temperature and its volume. A Partial Derivative measures how a function changes with respect to one variable while holding all others constant.
Notation: \(\frac{\partial f}{\partial x}\) (the "curly d").
Worked Examples
Example 1: Basic Partial
Find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for \(f(x, y) = x^2 y + 3y^3\).
- For \(\frac{\partial f}{\partial x}\): Treat \(y\) as a constant. \(\frac{\partial}{\partial x}(x^2 y) = 2xy\). \(\frac{\partial}{\partial x}(3y^3) = 0\). Result: \(2xy\).
- For \(\frac{\partial f}{\partial y}\): Treat \(x\) as a constant. \(\frac{\partial}{\partial y}(x^2 y) = x^2\). \(\frac{\partial}{\partial y}(3y^3) = 9y^2\). Result: \(x^2 + 9y^2\).
The Bridge to Quantum Mechanics
Particles in our universe move in three dimensions (\(x, y, z\)). The wavefunction is \(\psi(x, y, z, t)\). To find the kinetic energy in 3D, we must take the partial derivatives with respect to each coordinate: \(E_k = -\frac{\hbar^2}{2m} \left( \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} \right)\). This sum of second partial derivatives is called the Laplacian, and it is the heart of the 3D Schrödinger Equation.