Final answer:
To find the directional derivative of the function Φ(x, y) = e^(-x²-y²) at the point (3, 4) in the direction of the point (6, 7), we differentiate Φ(x, y) with respect to x and y, substitute the values (3, 4) into the derivative expressions, and evaluate the dot product.
Step-by-step explanation:
To find the directional derivative of the function Φ(x, y) = e-x²-y² at the point (3, 4) in the direction of the point (6, 7), we can use the formula:
∇Φ(3, 4) · (6, 7) = (∂Φ/∂x, ∂Φ/∂y) · (6, 7)
To calculate the partial derivatives, we differentiate Φ(x, y) with respect to x and y:
∂Φ/∂x = -2xe-x²-y²
∂Φ/∂y = -2ye-x²-y²
Substituting the values (3, 4) into the partial derivative expressions and then evaluating the dot product will give us the directional derivative.
∇Φ(3, 4) · (6, 7) = (-6e-x²-y², -8e-x²-y²) · (6, 7)
= (-6e-25, -8e-25) · (6, 7)
= -6e-25(6) - 8e-25(7)
= -92e-25