Answered: "Find the directional derivative of f at the..."

Sinead Mcgee

Answered question

2021-05-19

Find the directional derivative of f at the given point in the direction indicated by the angle

θ . f (x, y) = e^{x} \cos y, (0, 0), θ = \frac{π}{4}

Daphne Broadhurst

Skilled2021-05-20Added 109 answers

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Jeffrey Jordon

Expert2021-09-29Added 2605 answers

We need to find directional derivative of at the given point in the direction of v

$f (x, y) = e^{x} \cos y; θ = π / 4, (0, 0)$

Formula:

$D_{u} f (x, y) = f_{x} (x, y) a + f_{y} (x, y) b$

Find partial derivatives

$f_{x} = e^{x} \cos y$ and $f_{y} = - e^{x} \sin y$

Plug into formula

$D_{u} f = (e^{x} \cos y) \cos (π / 4) + (- e^{x} \sin y) \sin (π / 4)$

$= \frac{\sqrt{2}}{2} [(e^{x} \cos y) + (- e^{x} \sin (y)]$

Plug in values

$\frac{\sqrt{2}}{2} [e^{0} \cos 0 - e^{0} \sin 0]$

Result: $D_{u} f (0, 0) = \frac{\sqrt{2}}{2}$

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