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To solve the derivative of the given function \( y = 7t^2 + 6e^t \), we will use basic rules of differentiation:

1. For \( 7t^2 \), we use the power rule: \(\frac{d}{dt}(t^n) = n t^{n-1}\).
2. For \( 6e^t \), we use the fact that \(\frac{d}{dt}(e^t) = e^t\).

### Step-by-step solution:

Given:
\[
y = 7t^2 + 6e^t
\]

#### Derivative of the first term:
\[
\frac{d}{dt}(7t^2) = 7 \cdot 2t = 14t
\]

#### Derivative of the second term:
\[
\frac{d}{dt}(6e^t) = 6 \cdot e^t = 6e^t
\]

Thus, the derivative is:
\[
y' = 14t + 6e^t
\]

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to further your understanding:
1. How would the derivative change if we had \( y = 7t^3 + 6e^t \)?
2. What happens when you differentiate a product of two functions?
3. How do you differentiate the function \( e^{kt} \), where \( k \) is a constant?
4. What is the derivative of \( \ln(t) \)?
5. How do you differentiate trigonometric functions like \( \sin(t) \)?

**Tip:** Always apply the power rule carefully, and remember that the derivative of \( e^t \) remains \( e^t \).

Find the derivative of the function y = 7t^2 + 6e^t.

Learn how to find the derivative of the function y = 7t^2 + 6e^t using basic differentiation rules, including the power rule and the exponential function derivative. Step-by-step explanation provided. Suitable for students in Grades 11-12.

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