Let's go step by step through each problem to solve them.

Problem 1: Find the derivative of $y = - \frac{8}{x^2}$

We can rewrite this as:

$y = -8x^{-2}$

Now, apply the power rule of differentiation:

$\frac{dy}{dx} = \frac{d}{dx} [-8x^{-2}] = -8 \times (-2) x^{-3} = 16x^{-3} = \frac{16}{x^3}$

Problem 2: Calculate the derivative of $5t^{\frac{3}{2}}$

Use the power rule for $t^{\frac{3}{2}}$:

$\frac{d}{dt} \left( 5t^{\frac{3}{2}} \right) = 5 \times \frac{3}{2} t^{\frac{1}{2}} = \frac{15}{2} t^{\frac{1}{2}} = \frac{15}{2} \sqrt{t}$

Problem 3: Find the derivative of $f(x) = (x^2 + x)^{100}$

This requires the chain rule. First, let:

$u = x^2 + x$

So, $f(x) = u^{100}$. Using the chain rule:

$\frac{df}{dx} = 100u^{99} \cdot \frac{du}{dx}$

Now, differentiate $u = x^2 + x$:

$\frac{du}{dx} = 2x + 1$

Therefore:

$\frac{df}{dx} = 100(x^2 + x)^{99} \cdot (2x + 1)$

Problem 4: Find $\frac{dy}{dt}$ of $s(t) = \frac{t^3 + 7t + 1}{t}$

First, simplify the function:

$s(t) = t^2 + 7 + \frac{1}{t}$

Now, differentiate each term:

$\frac{ds}{dt} = 2t + 0 - t^{-2} = 2t - \frac{1}{t^2}$

Summary of Solutions:

1. $\frac{dy}{dx} = \frac{16}{x^3}$
2. $\frac{d}{dt} = \frac{15}{2} \sqrt{t}$
3. $\frac{df}{dx} = 100(x^2 + x)^{99} (2x + 1)$
4. $\frac{ds}{dt} = 2t - \frac{1}{t^2}$

Let me know if you would like more details or further clarifications.

Related Questions:

1. How does the power rule apply when differentiating functions of $x$?
2. Can you apply the product rule to functions similar to problem 3?
3. How can the quotient rule be used for more complex rational functions?
4. How does the chain rule work for composite functions?
5. What are the applications of derivatives in physics and engineering?

Tip: Always simplify expressions before differentiating—it makes the process smoother!