5 Questions
What is the notation for the mixed partial derivative of f with respect to y and then x?
$f_{yx}$
What is the notation for the second partial derivative of f with respect to x?
$f_{xx}$
What is the notation for the second partial derivative of f with respect to y?
$f_{yy}$
What is the notation for the mixed partial derivative of f with respect to x and then y?
$f_{xy}$
Under what condition are $f_{s}^{s-}$ and $f_{m}$ equal?
When $f_{s}^{s-}$ and $f_{m}$ exist
Study Notes
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.
Test your understanding of partial derivatives with this quiz. Explore the concept of partial derivatives of a real valued function defined on a subset of R², and how different types of limits yield different partial derivatives.
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