To find the derivative of the function $f(t) = 7t \sin(\pi t)$, we will use the product rule and chain rule.

Step 1: Apply the Product Rule

The product rule states: $(fg)' = f' g + f g'$ Here, we treat $f(t) = 7t$ and $g(t) = \sin(\pi t)$. Now, we differentiate each part separately:

1. Derivative of $7t$: $\frac{d}{dt}[7t] = 7$

2. Derivative of $\sin(\pi t)$ using the chain rule: [ \frac{d}{dt}[\sin(\pi t)] = \cos(\pi t) \cdot \frac{d}{dt}[\pi t] = \pi \cos(\pi t) ]

Step 2: Combine Using the Product Rule

Now apply the product rule: $f'(t) = (7t)' \cdot \sin(\pi t) + 7t \cdot (\sin(\pi t))'$ Substitute the derivatives we found: $f'(t) = 7 \cdot \sin(\pi t) + 7t \cdot \pi \cos(\pi t)$

Thus, the derivative of the function is: $f'(t) = 7 \sin(\pi t) + 7\pi t \cos(\pi t)$

Would you like a detailed breakdown of this result or have any further questions?

Here are 5 related questions to deepen your understanding:

1. How would you differentiate $f(t) = t^2 \sin(\pi t)$?
2. What is the derivative of $g(t) = e^t \cos(\pi t)$?
3. How does the product rule apply to three functions, say $f(t) = uvw$?
4. How does the chain rule work when differentiating $\sin^2(\pi t)$?
5. How can you find the second derivative of $f(t) = 7t \sin(\pi t)$?

Tip: The chain rule is essential when dealing with composite functions—always differentiate the outer function and then the inner function.