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We are tasked with finding the equation of the tangent line to the function \( f(x) = x^4 - 5x \) at the point \( x = 2 \), in point-slope form. The point-slope form of a line is given by:

\[
y - y_0 = m(x - x_0)
\]

Where:
- \( (x_0, y_0) \) is the point of interest, and
- \( m \) is the slope of the tangent line.

### Step 1: Find \( x_0 \) and \( y_0 \)
- \( x_0 \) is given as 2 (since we are asked for the tangent at \( x = 2 \)).
- To find \( y_0 \), plug \( x_0 = 2 \) into the function \( f(x) = x^4 - 5x \):

\[
y_0 = f(2) = 2^4 - 5(2) = 16 - 10 = 6
\]

So, \( y_0 = 6 \).

### Step 2: Find the slope \( m \)
The slope of the tangent line is the derivative of \( f(x) \) evaluated at \( x = 2 \). First, compute the derivative of \( f(x) \):

\[
f'(x) = \frac{d}{dx}(x^4 - 5x) = 4x^3 - 5
\]

Now, evaluate \( f'(x) \) at \( x = 2 \):

\[
m = f'(2) = 4(2^3) - 5 = 4(8) - 5 = 32 - 5 = 27
\]

### Step 3: Final Values
- \( x_0 = 2 \)
- \( y_0 = 6 \)
- \( m = 27 \)

Thus, the equation of the tangent line is:

\[
y - 6 = 27(x - 2)
\]

Let me know if you need more details or have any questions!

Here are five follow-up questions:
1. How would the tangent line change if the point was at \( x = 1 \) instead of \( x = 2 \)?
2. What is the geometric interpretation of the derivative \( f'(x) \)?
3. How can we find the second derivative of \( f(x) \), and what does it tell us about the function?
4. How would you use the chain rule if the function was more complex, like \( f(x) = (x^2 - 5)^2 \)?
5. Can we calculate the equation of the tangent line using different forms (e.g., slope-intercept)?

**Tip:** The slope of a tangent line gives the instantaneous rate of change of the function at a specific point.

Suppose f(x) = x^4 - 5x. If we were to express the equation of the tangent line to f(x) at x = 2 in point-slope form, what would be the following values?

Learn how to find the equation of the tangent line to the function f(x) = x^4 - 5x at the point x = 2. This step-by-step solution covers the derivative and point-slope form of a line, essential for understanding tangent lines in calculus.

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