To find the equation of the tangent line to the function \( f(x) = x \sqrt{x} \) at the point where \( x = 4 \), follow these steps:

### 1. **Find the function and point of tangency:**

We are given:
\[
f(x) = x \sqrt{x} = x \cdot x^{1/2} = x^{3/2}
\]

At \( x = 4 \), we can find the corresponding \( y \)-value:
\[
f(4) = 4 \cdot \sqrt{4} = 4 \cdot 2 = 8
\]

So, the point of tangency is \( (4, 8) \).

### 2. **Find the derivative:**

The derivative of the function will give us the slope of the tangent line. Differentiate \( f(x) = x^{3/2} \) with respect to \( x \):

\[
f'(x) = \frac{d}{dx} \left( x^{3/2} \right) = \frac{3}{2} x^{1/2}
\]

### 3. **Evaluate the derivative at \( x = 4 \):**

Now, find the slope of the tangent line by substituting \( x = 4 \) into the derivative:

\[
f'(4) = \frac{3}{2} \cdot \sqrt{4} = \frac{3}{2} \cdot 2 = 3
\]

So, the slope of the tangent line at \( x = 4 \) is \( 3 \).

### 4. **Write the equation of the tangent line:**

The equation of the tangent line is given by the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope, and \( (x_1, y_1) \) is the point of tangency.

Substituting \( m = 3 \), \( x_1 = 4 \), and \( y_1 = 8 \):
\[
y - 8 = 3(x - 4)
\]

Simplifying this:
\[
y - 8 = 3x - 12
\]
\[
y = 3x - 4
\]

### Final Answer:
The equation of the tangent line to \( f(x) = x \sqrt{x} \) at \( x = 4 \) is:
\[
y = 3x - 4
\]

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Would you like further details on any step or have any questions?

Here are some related questions you might consider:
1. What is the general approach for finding a tangent line to any function?
2. How does the derivative relate to the slope of a tangent line?
3. Can you find the tangent line at \( x = 1 \) for \( f(x) = x \sqrt{x} \)?
4. How does the point-slope form of a line work in tangent line problems?
5. What is the geometric interpretation of a tangent line?

**Tip:** The derivative gives the instantaneous rate of change, or slope, of the function at a specific point.

Learn how to find the tangent line to the function f(x) = x * sqrt(x) at x = 4. This step-by-step guide explains key calculus concepts such as derivatives and point-slope form, suitable for students in Grades 11-12 studying calculus.

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