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To find the derivative \( g'(t) \) of the given function \( g(t) = -5t^6 e^t + 10 \sin t \cos t \), we need to use both the product rule and the chain rule.

First, let's recall the rules we need:
1. **Product Rule**: If \( f(t) = u(t)v(t) \), then \( f'(t) = u'(t)v(t) + u(t)v'(t) \).
2. **Chain Rule**: If \( f(t) = h(g(t)) \), then \( f'(t) = h'(g(t)) \cdot g'(t) \).
3. **Derivative of \( \sin t \cos t \)**: Use the product rule here as well.

### Step-by-Step Solution

1. **Differentiate \( -5t^6 e^t \)**:
   \[
   \frac{d}{dt}(-5t^6 e^t) = -5 \left( \frac{d}{dt}(t^6) \cdot e^t + t^6 \cdot \frac{d}{dt}(e^t) \right)
   \]
   - \( \frac{d}{dt}(t^6) = 6t^5 \)
   - \( \frac{d}{dt}(e^t) = e^t \)

   Therefore:
   \[
   \frac{d}{dt}(-5t^6 e^t) = -5 \left( 6t^5 e^t + t^6 e^t \right) = -5 \left( 6t^5 + t^6 \right) e^t = -5 (6t^5 + t^6)e^t
   \]

2. **Differentiate \( 10 \sin t \cos t \)** using the product rule:
   \[
   \frac{d}{dt}(10 \sin t \cos t) = 10 \left( \frac{d}{dt}(\sin t) \cos t + \sin t \frac{d}{dt}(\cos t) \right)
   \]
   - \( \frac{d}{dt}(\sin t) = \cos t \)
   - \( \frac{d}{dt}(\cos t) = -\sin t \)

   Therefore:
   \[
   \frac{d}{dt}(10 \sin t \cos t) = 10 \left( \cos t \cdot \cos t + \sin t \cdot (-\sin t) \right) = 10 \left( \cos^2 t - \sin^2 t \right)
   \]

3. **Combine the results**:
   \[
   g'(t) = -5 (6t^5 + t^6)e^t + 10 (\cos^2 t - \sin^2 t)
   \]

Thus, the derivative \( g'(t) \) is:
\[
g'(t) = -5 (6t^5 + t^6)e^t + 10 (\cos^2 t - \sin^2 t)
\]

Would you like any further details or have any questions?

Here are 5 related questions that expand on this information:
1. How do you apply the product rule to functions with more than two factors?
2. What is the derivative of \( t^n e^{kt} \) for any constants \( n \) and \( k \)?
3. How do you differentiate trigonometric functions combined with exponential functions?
4. What is the importance of the chain rule in calculus?
5. How can we simplify the expression \( \cos^2 t - \sin^2 t \)?

**Tip:** Always write down each step when differentiating complex functions to avoid mistakes and better understand the process.

Compute the derivative of the given function g(t) = -5t^6 e^t + 10 sin t cos t

Learn how to compute the derivative of the function g(t) = -5t^6 e^t + 10 sin t cos t. This problem covers calculus concepts such as the product rule, chain rule, and differentiation of exponential and trigonometric functions.

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