4. What is the directional derivative of f(x,y) = 5 – 4x2 – 3y at (x,y) toward (0,0)?
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A. –8x – 3 |
B. (-8x2–3y)/ Ö (x2+y2) |
C. (-8x-3)/ Ö (64x2+9) |
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D. 8x2 + 3y |
E. (8x2+3y)/ Ö (x2+y2) |
Solution: The answer is E
The directional derivative is, by definition, the slope of the tangent line to the curve of intersection of the surface f(x,y) and the vertical plane through (x,y,0) in the direction of u. Where u is a unit vector formed from the directional vector v. The directional derivative in the direction of u, denoted Duf(x,y), is defined as,
Which, expresses the directional derivative as the scalar projection of the gradient vector onto u. The directional vector v is defined as,
Hence,
Calculate fx(x,y) and fy(x,y) as follows,
Combine the above results to calculate the directional derivative in the direction of u.
