Find 

gradient f∇f 

at the given point.

​f(x,y,z)equals=x squared plus y squared minus 3 z squared plus z ln xx2+y2−3z2+zlnx​, 

    left parenthesis 1 comma 5 comma 3 right parenthesis

To find the gradient of the function \( f(x, y, z) \), we first need to compute the partial derivatives of \( f \) with respect to each variable \( x \), \( y \), and \( z \).

### Function:
\[
f(x, y, z) = x^2 + y^2 - 3z^2 + z \ln x
\]

### Step 1: Compute the partial derivatives
1. **Partial derivative of \( f \) with respect to \( x \)**:
   \[
   \frac{\partial f}{\partial x} = 2x + \frac{z}{x}
   \]

2. **Partial derivative of \( f \) with respect to \( y \)**:
   \[
   \frac{\partial f}{\partial y} = 2y
   \]

3. **Partial derivative of \( f \) with respect to \( z \)**:
   \[
   \frac{\partial f}{\partial z} = -6z + \ln x
   \]

### Step 2: Evaluate the gradient at the point \( (1, 5, 3) \)

Substitute \( x = 1 \), \( y = 5 \), and \( z = 3 \) into the partial derivatives:

1. **For \( \frac{\partial f}{\partial x} \) at \( (1, 5, 3) \)**:
   \[
   \frac{\partial f}{\partial x} = 2(1) + \frac{3}{1} = 2 + 3 = 5
   \]

2. **For \( \frac{\partial f}{\partial y} \) at \( (1, 5, 3) \)**:
   \[
   \frac{\partial f}{\partial y} = 2(5) = 10
   \]

3. **For \( \frac{\partial f}{\partial z} \) at \( (1, 5, 3) \)**:
   \[
   \frac{\partial f}{\partial z} = -6(3) + \ln(1) = -18 + 0 = -18
   \]

### Step 3: Gradient at the point \( (1, 5, 3) \)

The gradient of \( f(x, y, z) \) at \( (1, 5, 3) \) is:

\[
\boxed{\nabla f = (5, 10, -18)}
\]

Would you like more details on any specific part of the solution, or do you have any questions?

Here are 5 related questions:
1. How do we interpret the gradient geometrically?
2. What does the gradient tell us about the function at the given point?
3. How would the gradient change if the point \( (x, y, z) \) were different?
4. Can you compute the directional derivative of \( f \) in a given direction at this point?
5. What is the second derivative (Hessian matrix) of \( f(x, y, z) \)?

**Tip:** The gradient vector points in the direction of the steepest ascent of a function at a given point.

Find the gradient ∇f at the given point: f(x,y,z) = x^2 + y^2 - 3z^2 + z ln(x), (1,5,3)

Discover how to calculate the gradient of the function f(x, y, z) = x^2 + y^2 - 3z^2 + z ln(x) at the point (1, 5, 3). This detailed step-by-step solution covers partial derivatives and key calculus concepts, suitable for college-level students.

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