Find the directional derivative of 𝑓(𝑥,𝑦)=𝑥𝑒^(4𝑦)−𝑦𝑒^(3𝑥) in the direction of 𝑣⃗ =⟨−2,3⟩ at the point 𝑃=(2,2).

Answer 1

To find a directional derivative, we find the gradient at our point of interest, then take the dot product of that gradient and the direction we're going. For this problem:

∇f=<∂f/∂x, ∂f/∂y> = <e^4y-3ye^3x, 4xe^4y-e^3x>

At point (2,2), ∇f=<e⁸-6e⁶, 8e⁸-e⁶>

Now we take ∇f•v = ( e⁸-6e⁶)*-2 +(8e⁸-e⁶)*3 = 6e⁸ + 11e⁶, and that's our answer!

I hope this made sense, good luck in your Calculus class, and don't be afraid to message me if you need any more help!