To find the instantaneous rate of change of the function $f(t) = \frac{5}{t} - 1$ at $t = -1$, we need to compute the derivative $f'(t)$, then evaluate it at $t = -1$.

Step 1: Differentiate $f(t)$

We use the power rule and the derivative of a constant to differentiate the function.

$f(t) = \frac{5}{t} - 1 = 5t^{-1} - 1$

The derivative of $5t^{-1}$ is:

$f'(t) = -5t^{-2}$

The derivative of a constant $-1$ is zero, so we have:

$f'(t) = -\frac{5}{t^2}$

Step 2: Evaluate the derivative at $t = -1$

Now, substitute $t = -1$ into the derivative:

$f'(-1) = -\frac{5}{(-1)^2} = -\frac{5}{1} = -5$

Thus, the instantaneous rate of change of the function at $t = -1$ is $-5$.

Summary:

The instantaneous rate of change of $f(t) = \frac{5}{t} - 1$ at $t = -1$ is $-5$.

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Would you like more details on the differentiation process, or do you have any questions?

Here are 5 related questions you may find helpful:

1. How do you compute the average rate of change of a function between two points?
2. What is the physical meaning of an instantaneous rate of change?
3. How do you interpret the derivative in real-life contexts?
4. Can you find the instantaneous rate of change for more complex functions?
5. How does the power rule work in derivative calculations?

Tip: When taking the derivative of a function involving a fraction, rewriting it with a negative exponent often makes differentiation easier!