To find the first derivative of the function $f(x) = 7x^2 e^x$, we will use the product rule. The product rule states that if you have a function $f(x) = u(x) \cdot v(x)$, then its derivative $f'(x)$ is given by:

$f'(x) = u'(x) v(x) + u(x) v'(x)$

In this case:

• $u(x) = 7x^2$
• $v(x) = e^x$

First, we find the derivatives of $u(x)$ and $v(x)$:

• $u'(x) = \frac{d}{dx}(7x^2) = 14x$
• $v'(x) = \frac{d}{dx}(e^x) = e^x$

Now, applying the product rule: $f'(x) = u'(x) v(x) + u(x) v'(x)$ $f'(x) = (14x)e^x + (7x^2)e^x$

We can factor out the common term $e^x$: $f'(x) = e^x (14x + 7x^2)$

So, the first derivative of the function $f(x) = 7x^2 e^x$ is: $f'(x) = e^x (14x + 7x^2)$

Do you want details or have any questions?

1. How is the product rule applied in differentiation?
2. What are the derivatives of basic exponential functions?
3. Why is factoring out common terms useful in derivatives?
4. What steps should you follow to verify your derivative calculations?
5. How does the product rule differ from the chain rule?
6. What are some common mistakes to avoid when using the product rule?
7. Can you explain the importance of understanding the derivatives of exponential functions?
8. What are the next steps after finding the first derivative for analyzing a function?

Tip: Always simplify your derivative expressions as much as possible to make further calculations easier.