Consider the following.

*yz* = 5 ln(*x* + *z*),   (0, 0, 1)

(a)

Find an equation of the tangent plane to the given surface at the specified point.

(b)

Find an equation of the normal line to the given surface at the specified point.

To solve this problem, we will find the equation of the tangent plane and the equation of the normal line to the given surface at the specified point \((x_0, y_0, z_0)\).

### Step 1: Restate the problem

We are given the surface:

\[
f(x, y, z) = yz - 5 \ln(x + z)
\]

At the point \((0, 0, 1)\), we need to:
1. Find the equation of the **tangent plane**.
2. Find the equation of the **normal line**.

### Step 2: Partial Derivatives for the Tangent Plane

To find the tangent plane, we first need the gradient of \(f(x, y, z)\). This is calculated using partial derivatives with respect to \(x\), \(y\), and \(z\).

#### Partial derivative with respect to \(x\):

\[
f_x(x, y, z) = \frac{\partial}{\partial x} \left( yz - 5 \ln(x + z

Consider the following. yz = 5 ln(x + z), (0, 0, 1)
(a) Find an equation of the tangent plane to the given surface at the specified point.
(b) Find an equation of the normal line to the given surface at the specified point.

Learn how to find the tangent plane and normal line to the surface yz = 5 ln(x + z) at the point (0, 0, 1). This problem covers concepts from multivariable calculus, including partial derivatives, the gradient vector, and the equations for tangent planes and normal lines. Suitable for undergraduate calculus students.

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