Professor Leonard – Calculus 3 – Lecture 13.3: Partial Derivatives (Derivatives of Multivariable Functions)
Thomas – 14.3 Partial Derivatives
The Partial Derivative
A function f(x, y) is represented by a surface in \mathbb{R}^3 . In single-variable calculus, the derivative is the slope of the tangent line to a curve at a point. A surface, however, has infinitely many tangent directions through each point. As such, there are infinitely many possible directions, so there is no single derivative at a point.
To define a derivative, we first choose a direction in which to measure the rate of change. The simplest way to do this is to choose directions which run parallel to the x and y-axes. This is done by holding one variable constant while allowing the other to vary. Geometrically, this is like slicing the surface with a plane parallel to either the xz-plane or yz-plane. The intersection of the surface and the plane forms a curve. The resulting slice is a single-variable function which can be differentiated exactly as was learned in early Calculus I. The derivative obtained by holding all other variables constant is called a partial derivative.
Notation
The partial derivative with respect to x is denoted
- f_x
- \frac{\partial f}{\partial x}
- \partial_x f (less common)
The partial derivative with respect to y is denoted
- f_y
- \frac{\partial f}{\partial y}
How to Compute Partial Derivatives
To find f_x, treat every occurrence of y as a constant. To find f_y, treat every occurrence of x as a constant. Differentiate exactly as you would in single-variable calculus.