 Find the derivative of the function.
f(t) = 7t sin(𝜋t)


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.

Find the derivative of the function. f(t) = 7t sin(𝜋t)

Learn how to find the derivative of the function f(t) = 7t sin(πt) using the product rule and chain rule. This solution covers advanced differentiation techniques and applies trigonometric function derivatives. Suitable for students in Grades 11-12.

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