Let 
f(t) = 
3

t
.
For 
a ≠ 0,
 find 
f ′(a).

We are tasked with finding the derivative of the function \( f(t) = \frac{3}{t} \) at \( t = a \), where \( a \neq 0 \).

### Step 1: Rewrite the function
The given function is:
\[
f(t) = \frac{3}{t}
\]
This can also be written as:
\[
f(t) = 3t^{-1}
\]

### Step 2: Differentiate the function
Now we apply the power rule for differentiation. The power rule states that \( \frac{d}{dt}(t^n) = n t^{n-1} \).

For \( f(t) = 3t^{-1} \):
\[
f'(t) = 3 \cdot (-1) \cdot t^{-2}
\]
\[
f'(t) = -3t^{-2}
\]
This is the derivative of \( f(t) \) with respect to \( t \).

### Step 3: Evaluate the derivative at \( t = a \)
Now, substitute \( t = a \) into \( f'(t) \):
\[
f'(a) = -3a^{-2}
\]
Thus, the derivative of the function at \( t = a \) is:
\[
f'(a) = -\frac{3}{a^2}
\]

### Final Answer:
\[
f'(a) = -\frac{3}{a^2}
\]

Would you like more details or further explanation?

Here are 5 related questions you might find helpful:
1. What is the power rule of differentiation and how does it apply?
2. How can you find the second derivative of \( f(t) = \frac{3}{t} \)?
3. What happens to the derivative when \( t = 0 \)?
4. How do derivatives of rational functions generally behave?
5. How does differentiation handle negative exponents?

**Tip:** Always simplify functions before differentiating when possible. This often makes applying rules easier.

Let f(t) = 3/t. For a ≠ 0, find f ′(a).

Learn how to differentiate f(t) = 3/t at t = a using the power rule for derivatives. This solution breaks down the process step-by-step and is perfect for calculus students in grades 11-12 or early college.

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