Find the derivative of y with respect to x if y equals left parenthesis 3 x squared minus 6 x plus 6 right parenthesis e Superscript negative 5 x.

We are tasked with finding the derivative of:

\[
y = (3x^2 - 6x + 6)e^{-5x}
\]

This is a product of two functions: \( u(x) = 3x^2 - 6x + 6 \) and \( v(x) = e^{-5x} \). To differentiate this, we will apply the **product rule**:

\[
\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
\]

### Step 1: Differentiate \( u(x) \)
\[
u(x) = 3x^2 - 6x + 6
\]
\[
u'(x) = \frac{d}{dx}(3x^2 - 6x + 6) = 6x - 6
\]

### Step 2: Differentiate \( v(x) \)
\[
v(x) = e^{-5x}
\]
\[
v'(x) = \frac{d}{dx}(e^{-5x}) = -5e^{-5x}
\]

### Step 3: Apply the product rule
Now, applying the product rule:

\[
\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)
\]

Substituting the derivatives:

\[
\frac{dy}{dx} = (6x - 6)e^{-5x} + (3x^2 - 6x + 6)(-5e^{-5x})
\]

### Step 4: Simplify the expression
Factor out \( e^{-5x} \):

\[
\frac{dy}{dx} = e^{-5x} \left[ (6x - 6) + (-5)(3x^2 - 6x + 6) \right]
\]

Expand the second term:

\[
= e^{-5x} \left[ 6x - 6 - 15x^2 + 30x - 30 \right]
\]

Combine like terms:

\[
= e^{-5x} \left[ -15x^2 + 36x - 36 \right]
\]

Thus, the derivative is:

\[
\frac{dy}{dx} = e^{-5x}(-15x^2 + 36x - 36)
\]

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

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Here are 5 related questions:
1. How does the product rule apply to other exponential functions?
2. What is the significance of negative exponents in derivatives?
3. How do you factor a quadratic function like \( -15x^2 + 36x - 36 \)?
4. How can you check if the derivative is correct using limits?
5. What are some real-world applications of exponential derivatives?

**Tip**: When using the product rule, always differentiate each function separately before combining them!

This step-by-step guide demonstrates how to find the derivative of y = (3x^2 - 6x + 6)e^(-5x) using the product rule and properties of exponential functions. The solution involves differentiating each function separately and applying the product rule. Suitable for high school or college calculus students.

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