Question

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

f(x,y)=x³y⁴ +x⁴y³ ,(1,1),

Answer 1

The directional derivative of the function f(x,y) = x³y⁴ + x⁴y³ at the point (1,1) is found by first calculating the gradient at that point, then determining the corresponding unit vector for the given direction, and finally taking the dot product of the gradient and the unit vector.

Step-by-step explanation:

You are asked to find the directional derivative of the function f(x,y) = x³y⁴ + x⁴y³ at the point (1,1) in the direction indicated by a certain angle. To find the directional derivative, you need to calculate the gradient of the function and then use it to find the derivative in the direction of the given angle.

The gradient is a vector containing the partial derivatives of f with respect to x and y. First, calculate the partial derivatives at the point (1,1). Then, determine the unit vector in the direction of the given angle. The directional derivative is the dot product of the gradient and the unit vector corresponding to the angle.

To calculate the gradient, apply the following steps:

1. Find the partial derivative of f with respect to x, which is fx.
2. Find the partial derivative of f with respect to y, which is fy.
3. Evaluate both derivatives at the point (1,1).

For the direction, you need the angle in standard position, convert the given direction angle to a unit vector in Cartesian coordinates.

The directional derivative is then calculated as the dot product of the gradient of f at (1,1) and the unit direction vector.