First and second derivative of (t^2 - 1)/t

To find the first and second derivatives of the function \( f(t) = \frac{t^2 - 1}{t} \), we'll start by simplifying the function and then differentiate it.

### Simplifying the Function

\[
f(t) = \frac{t^2 - 1}{t}
\]

We can simplify this by dividing each term in the numerator by \( t \):

\[
f(t) = \frac{t^2}{t} - \frac{1}{t} = t - \frac{1}{t}
\]

### First Derivative

Now, let's find the first derivative \( f'(t) \):

\[
f'(t) = \frac{d}{dt}\left( t - \frac{1}{t} \right)
\]

1. The derivative of \( t \) with respect to \( t \) is \( 1 \).
2. The derivative of \( \frac{1}{t} \) is \( -\frac{1}{t^2} \).

So,

\[
f'(t) = 1 + \frac{1}{t^2}
\]

### Second Derivative

Next, we'll find the second derivative \( f''(t) \):

\[
f''(t) = \frac{d}{dt}\left( 1 + \frac{1}{t^2} \right)
\]

1. The derivative of \( 1 \) is \( 0 \).
2. The derivative of \( \frac{1}{t^2} \) is \( -\frac{2}{t^3} \).

So,

\[
f''(t) = -\frac{2}{t^3}
\]

### Summary

- **First derivative**: \( f'(t) = 1 + \frac{1}{t^2} \)
- **Second derivative**: \( f''(t) = -\frac{2}{t^3} \)

Would you like any further details or explanations?

Here are five related questions to consider:
1. What is the first derivative of \( \frac{2t^3 - t^2 + 1}{t} \)?
2. How do you find the second derivative of \( \frac{t^2 + 2t + 1}{t} \)?
3. Can you find the critical points of \( f(t) = \frac{t^2 - 1}{t} \) by using the first derivative?
4. What happens to the first derivative \( f'(t) \) as \( t \) approaches infinity?
5. How do you find the concavity of a function using the second derivative?

**Tip:** When dealing with derivatives of rational functions, always consider simplifying the function first; it can make the differentiation process much easier!

Learn how to find the first and second derivatives of the function f(t) = (t^2 - 1)/t. This problem involves differentiation of rational functions, covering rules for derivatives and simplification techniques. Suitable for Grades 11-12.

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