Question

The directional derivative of a function \( f(x, y, z) = 4x^2 + 8y^2 + 9z^2 \) at the point \( P(3, 4, 5) \) in the direction vector \( \vec{b} = 2\hat{i} - 3\hat{j} + 4\hat{k} \) is _______ (rounded off to 1 decimal place).

Hint:

The directional derivative measures the rate at which a function changes in a specified direction. It is computed as the dot product of the gradient and the unit direction vector.

Answer 1

Step 1: Compute the gradient of \( f(x, y, z) \). \[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = \left( 8x, 16y, 18z \right) \] At point \( P(3, 4, 5) \), \[ \nabla f = \langle 24, 64, 90 \rangle \] Step 2: Normalize the direction vector \( \vec{b} = \langle 2, -3, 4 \rangle \). \[ |\vec{b}| = \sqrt{2^2 + (-3)^2 + 4^2} = \sqrt{4 + 9 + 16} = \sqrt{29} \] \[ \hat{u} = \left\langle \frac{2}{\sqrt{29}}, \frac{-3}{\sqrt{29}}, \frac{4}{\sqrt{29}} \right\rangle \] Step 3: Compute the directional derivative using dot product. \[ D_{\vec{b}}f = \nabla f \cdot \hat{u} = \langle 24, 64, 90 \rangle \cdot \left\langle \frac{2}{\sqrt{29}}, \frac{-3}{\sqrt{29}}, \frac{4}{\sqrt{29}} \right\rangle \] \[ = \frac{1}{\sqrt{29}} (24 \cdot 2 + 64 \cdot (-3) + 90 \cdot 4) = \frac{1}{\sqrt{29}} (48 - 192 + 360) = \frac{216}{\sqrt{29}} \] \[ \sqrt{29} \approx 5.385 \quad \Rightarrow \quad \frac{216}{5.385} \approx 40.1 \]