Mathematics · University
Directional derivative along a unit vector
Normalize a direction vector before the dot product. Here (3/5,4/5) is already a unit vector.
Method and conditions
Normalize a direction vector before the dot product. Here (3/5,4/5) is already a unit vector.
Dᵤf = ∇f · u
Worked examples
For f(x,y) = 7x² + 9y², find the directional derivative at (0,-3) in the unit direction (3/5,4/5). Round the final result to 4 decimal places if needed.
The gradient at the point is (0, -54). Dot this with (3/5,4/5).
-43.2
For f(x,y) = 3x² + 9y², find the directional derivative at (6,-1) in the unit direction (3/5,4/5). Round the final result to 4 decimal places if needed.
The gradient at the point is (36, -18). Dot this with (3/5,4/5).
7.2
Try it
For f(x,y) = 10x² + 8y², find the directional derivative at (2,-1) in the unit direction (3/5,4/5). Round the final result to 4 decimal places if needed.
For f(x,y) = 11x² + 4y², find the directional derivative at (-3,-1) in the unit direction (3/5,4/5). Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 11.2
The gradient at the point is (40, -16). Dot this with (3/5,4/5).
- -46
The gradient at the point is (-66, -8). Dot this with (3/5,4/5).
Further reading and review references
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