Partial Derivatives Quiz

Partial Derivatives

• Partial derivatives play a crucial role in differential calculus.
• The concept of partial derivatives is based on the different ways of limits discussed in the previous section.

Definition of Partial Derivatives

• Consider a real-valued function f(x, y) defined on a subset E of R², where E contains a neighborhood of (a, b) in R².
• Let Δa be a change in a; if the limit lim f(a + Δa, b) - f(a, b) / Δa exists, then it is called the partial derivative of f with respect to x at (a, b).
• The partial derivative of f with respect to x at (a, b) is denoted by ∂f/∂x|{(a, b)} or fx(a, b) or zx(a, b).
• Similarly, let Δb be a change in b; if the limit lim f(a, b + Δb) - f(a, b) / Δb exists, then it is called the partial derivative of f with respect to y at (a, b).
• The partial derivative of f with respect to y at (a, b) is denoted by ∂f/∂y|{(a, b)} or fy(a, b) or zy(a, b).

Notations

• If the partial derivatives fx and fy exist at each point of E, then they are also real-valued functions on E.
• We can obtain the partial derivatives of these functions if they are differentiable.
• We use the following notations: ∂f/∂x, ∂f/∂y, fx, fy, zx, and zy.