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To solve this problem and find the equation of the tangent line to the graph of \( f(x) = -5xe^x \) at the point \( a = 2 \), we will follow these steps:

### Step 1: Compute the value of \( f(2) \)
We are given that \( f(x) = -5xe^x \), so to find the y-coordinate of the point \( (2, f(2)) \), substitute \( x = 2 \) into the function:
\[
f(2) = -5(2)e^2 = -10e^2
\]

### Step 2: Find \( f'(x) \) (the derivative of \( f(x) \))
To find the equation of the tangent line, we need the slope of the tangent at \( x = 2 \), which is given by \( f'(2) \). We will first compute the derivative \( f'(x) \) using the product rule:
\[
f(x) = -5x \cdot e^x
\]
The product rule is \( (uv)' = u'v + uv' \). Let \( u = -5x \) and \( v = e^x \), then:
- \( u' = -5 \)
- \( v' = e^x \)

Thus, applying the product rule:
\[
f'(x) = (-5)(e^x) + (-5x)(e^x) = -5e^x - 5xe^x
\]

### Step 3: Compute \( f'(2) \)
Now, substitute \( x = 2 \) into \( f'(x) \) to get the slope at \( x = 2 \):
\[
f'(2) = -5e^2 - 5(2)e^2 = -5e^2 - 10e^2 = -15e^2
\]
So, the slope of the tangent line is \( -15e^2 \).

### Step 4: Use the point-slope form of the equation of a line
The equation of the tangent line can be written using the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency. Here:
- \( m = -15e^2 \)
- \( (x_1, y_1) = (2, -10e^2) \)

Substitute these values into the point-slope equation:
\[
y - (-10e^2) = -15e^2(x - 2)
\]
Simplify the equation:
\[
y + 10e^2 = -15e^2(x - 2)
\]
\[
y = -15e^2(x - 2) - 10e^2
\]

This is the equation of the tangent line.

Would you like further details on any of the steps? Here are five related questions to explore further:

1. How does the product rule help when differentiating a product of two functions?
2. How is the point-slope form of a line derived?
3. What other methods can be used to find the slope of a curve at a given point?
4. What is the significance of the tangent line to a function at a specific point?
5. How does the exponential function \( e^x \) behave as \( x \) increases?

**Tip**: Always verify your final result by plugging the point into both the original function and the derived tangent equation to ensure accuracy!

Find an equation for the line tangent to the graph of f(x) = -5xe^x at the point (a, f(a)) for a = 2.

Learn how to find the equation of the tangent line to the graph of f(x) = -5xe^x at x = 2. This solution walks through computing the derivative using the product rule and applying the point-slope form. Suitable for advanced calculus students.

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