Consider a vector $\overline{u}=2\hat{x}+\hat{y}+2\hat{z}$, where $\hat{x},\hat{y},\hat{z}$ represent unit vectors along the coordinate axes x, y, z respectively. The directional derivative of the function $f(x,y,z)=2\ln (xy)+\ln (yz)+3\ln (xz)$ at the point $(x,y,z)=(1,1,1)$ in the direction of $\overline{u}$ is :

The correct answer is 7.

To find the directional derivative of the function at a given point in the direction of a vector, we follow a systematic mathematical approach:

Step 1: Simplify the given function
The given function is:
$$f(x,y,z)=2\ln (xy)+\ln (yz)+3\ln (xz)$$
Using the logarithmic property $\ln (ab)=\ln a+\ln b$, we can expand each term:
$$f(x,y,z)=2(\ln x+\ln y)+(\ln y+\ln z)+3(\ln x+\ln z)$$
Now, group the terms for $\ln x$, $\ln y$, and $\ln z$:
$$f(x,y,z)=(2+3)\ln x+(2+1)\ln y+(1+3)\ln z$$
$$f(x,y,z)=5\ln x+3\ln y+4\ln z$$

Step 2: Find the Gradient of the function
The gradient vector $\nabla f$ is defined as:
$$\nabla f=\frac{\partial f}{\partial x}\hat{x}+\frac{\partial f}{\partial y}\hat{y}+\frac{\partial f}{\partial z}\hat{z}$$
Calculating the partial derivatives:
$$\frac{\partial f}{\partial x}=\frac{5}{x},\frac{\partial f}{\partial y}=\frac{3}{y},\frac{\partial f}{\partial z}=\frac{4}{z}$$
Thus, the gradient vector is:
$$\nabla f=\frac{5}{x}\hat{x}+\frac{3}{y}\hat{y}+\frac{4}{z}$$

Step 3: Evaluate the gradient at the point (1, 1, 1)
Substituting $x=1$, $y=1$, and $z=1$:
$$\nabla f{|}_{(1,1,1)}=5\hat{x}+3\hat{y}+4\hat{z}$$

Step 4: Find the unit vector in the direction of $\overline{u}$
The given vector is:
$$\overline{u}=2\hat{x}+\hat{y}+2\hat{z}$$
The magnitude of $\overline{u}$ is:
$$|\overline{u}|=\sqrt{2^2+1^2+2^2}=\sqrt{4+1+4}=\sqrt{9}=3$$
The unit vector $\hat{u}$ in the direction of $\overline{u}$ is:
$$\hat{u}=\frac{\overline{u}}{|\overline{u}|}=\frac{2\hat{x}+\hat{y}+2\hat{z}}{3}$$

Step 5: Calculate the Directional Derivative
The directional derivative is the dot product of the gradient at the point and the unit direction vector:
$$D_{\hat{u}}f=\nabla f·\hat{u}$$
Substitute the vectors:
$$D_{\hat{u}}f=(5\hat{x}+3\hat{y}+4\hat{z})·\left(\frac{2\hat{x}+\hat{y}+2\hat{z}}{3}\right)$$
Compute the dot product:
$$D_{\hat{u}}f=\frac{(5\times 2)+(3\times 1)+(4\times 2)}{3}$$
$$D_{\hat{u}}f=\frac{10+3+8}{3}=\frac{21}{3}=7$$