01 Key Concepts
Functions of Several Variables
A function like f(x,y) depends on two (or more) inputs, producing a surface rather than a curve.
Notation
Partial derivative with respect to x is written as (partial f / partial x) or f_x. Same idea applies for y, giving f_y.
How to Compute
To find f_x, differentiate f(x,y) with respect to x while treating y as a constant. To find f_y, do the reverse.
Geometric Meaning
f_x gives the slope of the surface in the x-direction at a point; f_y gives the slope in the y-direction.
Higher-Order Partials
You can take partial derivatives of partial derivatives, e.g., f_xx, f_yy, or the mixed partial f_xy.
02 Solved Examples
Example 1 Find the partial derivatives of f(x,y) = x^2*y + y^3.
- To find f_x, treat y as a constant: d/dx[x^2*y] = 2xy, and d/dx[y^3] = 0.
- So f_x = 2xy.
- To find f_y, treat x as a constant: d/dy[x^2*y] = x^2, and d/dy[y^3] = 3y^2.
- So f_y = x^2 + 3y^2.
Answer: f_x = 2xy, f_y = x^2 + 3y^2
Example 2 Find f_x and f_y for f(x,y) = 3x^2 + 4xy + y^2.
- f_x: treat y as constant. d/dx[3x^2]=6x, d/dx[4xy]=4y, d/dx[y^2]=0.
- So f_x = 6x + 4y.
- f_y: treat x as constant. d/dy[3x^2]=0, d/dy[4xy]=4x, d/dy[y^2]=2y.
- So f_y = 4x + 2y.
Answer: f_x = 6x + 4y, f_y = 4x + 2y
Example 3 Find f_xy (mixed partial) for f(x,y) = x^3*y^2.
- First find f_x = 3x^2*y^2 (treating y as constant).
- Now differentiate f_x with respect to y: d/dy[3x^2*y^2] = 6x^2*y.
Answer: f_xy = 6x^2*y
03 Practice Questions
1Find f_x for f(x,y) = x^2 + y^2.
2x
2Find f_y for f(x,y) = x^2 + y^2.
2y
3Find f_x for f(x,y) = 3xy + y^2.
3y
4Find f_y for f(x,y) = 3xy + y^2.
3x + 2y
5Find f_x for f(x,y) = x^2*y^3.
2xy^3
📄 Partial Derivatives — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.