Calculate the directional derivative of 𝑓(𝑥,𝑦,𝑧)=−4𝑥^2−5𝑦^2−3𝑧^2 at the point 𝑃=(4,−5,−4) in the direction of the origin.

Answer 1

Calculate the directional derivative of 𝑓(𝑥,𝑦,𝑧)=−4𝑥^2−5𝑦^2−3𝑧^2 at the point 𝑃=(4,−5,−4)

*-

-8xi-10yj-6zk

substitute

∇f(4,-5,-4)=-32i+50j+24k

Answer 2

The gradient is (-8x, -10y, -6z), which at (4, -5, -4) is (-32, 50, 24)

Then, in the direction (-4, 5, 4), this is (-4*-32+5*50+24*4)/√(4^2+5^2+4^2) = 474/√57, and the directional derivative toward (0,0, 0) is (-32*474,50*474,24*474)/57 = (-1896,2370,1896)/57 = (-632/19,790/19,632/19)