only 58 ## Finding a Directional DerivativeIn Exercises 57 and 58, use Theorem 13.9 to find the directional derivative of the function at \( P \) in the direction of \( \mathbf{v} \).### Exercise 57Given:- Function \( f(x, y) = x^2 y \)- Point \( P(-5, 5) \)- Direction vector \( \mathbf{v} = 3i - 4j \)### Exercise 58Given:- Function \( f(x, y) = \frac{1}{4}y^2 - x^2 \)- Point \( P(1, 4) \)- Direction vector \( \mathbf{v} = 2i + j \)#### Explanation of Theorem 13.9Theorem 13.9 typically involves using the gradient of the function \( \nabla f \) and the given direction vector \( \mathbf{v} \) to find the directional derivative. The directional derivative \( D_{\mathbf{v}} f \) at a point \( P \) is given by:\[ D_{\mathbf{v}} f = \nabla f \cdot \mathbf{u} \]where \( \mathbf{u} \) is the unit vector in the direction of \( \mathbf{v} \) (i.e., \( \mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} \)) and \( \cdot \) represents the dot product.#### Steps to Solve:1. **Compute the Gradient \( \nabla f \)**: Find the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \).2. **Evaluate the Gradient at Point \( P \)**: Substitute the coordinates of \( P \) into \( \nabla f \) to obtain \( \nabla f(P) \).3. **Normalize the Direction Vector \( \mathbf{v} \)**: Find the unit vector \( \mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} \).4. **Compute the Dot Product**: Find the dot product of \( \nabla f(P) \) and \( \mathbf{u} \).By following these steps, the directional derivatives for Exercises 57 and 58 can be calculated

Name: only 58 ## Finding a Directional DerivativeIn Exercises 57 and 58, use
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: only 58 ## Finding a Directional DerivativeIn Exercises 57 and 58, use Theorem 13.9 to find the directional derivative of the function at \( P \) in the direction of \( \mathbf{v} \).### Exercise 57Given:- Function \( f(x, y) = x^2 y \)- Point \( P(-