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Part 1
Find the derivative of the function.
y equals left parenthesis 13 x squared minus 26 x plus 26 right parenthesis e Superscript negative 5 x


To find the derivative of the given function \( y = (13x^2 - 26x + 26)e^{-5x} \), we will apply the **product rule**. The product rule states that if \( y = u(x)v(x) \), then the derivative of \( y \) is:

\[
y' = u'(x)v(x) + u(x)v'(x)
\]

### Step 1: Identify \( u(x) \) and \( v(x) \)

Let:
- \( u(x) = 13x^2 - 26x + 26 \)
- \( v(x) = e^{-5x} \)

### Step 2: Compute \( u'(x) \)

Differentiate \( u(x) = 13x^2 - 26x + 26 \):

\[
u'(x) = 26x - 26
\]

### Step 3: Compute \( v'(x) \)

Differentiate \( v(x) = e^{-5x} \). Using the chain rule, the derivative of \( e^{kx} \) is \( ke^{kx} \), so:

\[
v'(x) = -5e^{-5x}
\]

### Step 4: Apply the product rule

Now, apply the product rule:

\[
y' = u'(x)v(x) + u(x)v'(x)
\]

Substitute the expressions for \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \):

\[
y' = (26x - 26)e^{-5x} + (13x^2 - 26x + 26)(-5e^{-5x})
\]

### Step 5: Simplify the expression

Factor out \( e^{-5x} \):

\[
y' = e^{-5x} \left( (26x - 26) + (-5)(13x^2 - 26x + 26) \right)
\]

Simplify the terms inside the parentheses:

\[
y' = e^{-5x} \left( 26x - 26 - 65x^2 + 130x - 130 \right)
\]

\[
y' = e^{-5x} \left( -65x^2 + 156x - 156 \right)
\]

Thus, the derivative is:

\[
y' = e^{-5x}(-65x^2 + 156x - 156)
\]

### Final Answer:

\[
y' = e^{-5x}(-65x^2 + 156x - 156)
\]

---

Would you like more details on this? Here are some related questions you might find interesting:

1. How do we differentiate functions using the chain rule?
2. Can we use other rules to simplify the product of functions?
3. How does the chain rule apply to exponential functions?
4. What is the significance of \( e^{-5x} \) in this context?
5. What if the exponent of \( e^{-5x} \) were a more complicated function?

**Tip**: Always look for common factors after applying the product rule to make simplification easier.

Find the derivative of the function: y = (13x^2 - 26x + 26) e^{-5x}

Learn how to find the derivative of the function y = (13x^2 - 26x + 26) e^{-5x} by applying the product rule and chain rule. This problem covers key calculus concepts like the product rule and differentiation of exponential functions. Suitable for students in Grades 11-12 or early college-level calculus.

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