Find the directional derivative of the function at \( P \) in the direction of \( \mathbf{v} \). \[ h(x, y)=e^{4 x} \sin (9 y), \quad P\left(\frac{1}{4}, \frac{\pi}{18}\right), \mathbf{v}=-\mathbf{i} \]

To find the directional derivative of the function \( h(x, y) = e^{4x} \sin(9y) \) at the point \( P\left(\frac{1}{4}, \frac{\pi}{18}\right) \) in the direction of \( \mathbf{v} = -\mathbf{i} \), we first need to compute the gradient of the function. The gradient \( \nabla h \) is given by: \[ \nabla h = \left( \frac{\partial h}{\partial x}, \frac{\partial h}{\partial y} \right). \] Now we calculate the partial derivatives: 1. **Partial derivative with respect to \( x \)**: \[ \frac{\partial h}{\partial x} = 4 e^{4x} \sin(9y). \] 2. **Partial derivative with respect to \( y \)**: \[ \frac{\partial h}{\partial y} = 9 e^{4x} \cos(9y). \] Next, we evaluate the gradient at the point \( P \left(\frac{1}{4}, \frac{\pi}{18}\right) \): 1. Calculate \( \frac{\partial h}{\partial x} \) at \( P \): \[ \frac{\partial h}{\partial x}\left(\frac{1}{4}, \frac{\pi}{18}\right) = 4 e^{4 \cdot \frac{1}{4}} \sin\left(9 \cdot \frac{\pi}{18}\right) = 4 e^{1} \sin\left(\frac{\pi}{2}\right) = 4 e. \] 2. Calculate \( \frac{\partial h}{\partial y} \) at \( P \): \[ \frac{\partial h}{\partial y}\left(\frac{1}{4}, \frac{\pi}{18}\right) = 9 e^{4 \cdot \frac{1}{4}} \cos\left(9 \cdot \frac{\pi}{18}\right) = 9 e^{1} \cos\left(\frac{\pi}{2}\right) = 0. \] So the gradient at point \( P \) is: \[ \nabla h\left(\frac{1}{4}, \frac{\pi}{18}\right) = (4e, 0). \] The directional derivative \( D_{\mathbf{v}} h \) in the direction of \( \mathbf{v} \) is given by: \[ D_{\mathbf{v}} h = \nabla h \cdot \mathbf{u}, \] where \( \mathbf{u} \) is the unit vector in the direction of \( \mathbf{v} \). Since \( \mathbf{v} = -\mathbf{i} \), we have: \[ \mathbf{u} = \frac{\mathbf{v}}{||\mathbf{v}||} = \frac{-\mathbf{i}}{1} = (-1, 0). \] Now we compute the dot product: \[ D_{\mathbf{v}} h = (4e, 0) \cdot (-1, 0) = 4e \cdot (-1) + 0 \cdot 0 = -4e. \] Thus, the directional derivative of the function at point \( P \) in the direction of \( \mathbf{v} \) is: \[ \boxed{-4e}. \]