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Let's go through the problems one by one and solve them. The task is to find the derivatives of the given functions.

### Problem a: 
\[ f(x) = x^7 \]

To find the derivative of \( f(x) \), apply the power rule:

\[
f'(x) = 7x^{7-1} = 7x^6
\]

### Problem b:
\[ g(x) = \frac{1}{\sqrt{x}} \]

This can be rewritten as \( g(x) = x^{-\frac{1}{2}} \).

Now, apply the power rule:

\[
g'(x) = -\frac{1}{2}x^{-\frac{3}{2}}
\]

### Problem c:
\[ h(x) = x^2 + 5x - 7 \]

For each term, apply the power rule separately:

- Derivative of \( x^2 \) is \( 2x \),
- Derivative of \( 5x \) is \( 5 \),
- Derivative of \( -7 \) is 0 (since it's a constant).

Thus:
\[
h'(x) = 2x + 5
\]

### Problem d:
\[ y = 4x^3 - 3x\sqrt{x} + \frac{1}{2}x \]

First, rewrite the square root term \( \sqrt{x} \) as \( x^{1/2} \), so we get:
\[ y = 4x^3 - 3x^{3/2} + \frac{1}{2}x \]

Now, differentiate each term:

- \( 4x^3 \) gives \( 12x^2 \),
- \( -3x^{3/2} \) gives \( -\frac{9}{2}x^{1/2} \),
- \( \frac{1}{2}x \) gives \( \frac{1}{2} \).

Thus:
\[
y' = 12x^2 - \frac{9}{2}x^{1/2} + \frac{1}{2}
\]

### Problem e:
\[ h(t) = \frac{2}{t\sqrt{t}} - 3t \]

First, simplify \( \frac{2}{t\sqrt{t}} \). Since \( \sqrt{t} = t^{1/2} \), we have:
\[
\frac{2}{t\sqrt{t}} = \frac{2}{t^{3/2}} = 2t^{-3/2}
\]

Now differentiate:

- Derivative of \( 2t^{-3/2} \) is \( -3t^{-5/2} \),
- Derivative of \( -3t \) is \( -3 \).

Thus:
\[
h'(t) = -3t^{-5/2} - 3
\]

---

Let me know if you'd like further explanations or if you have any questions!

Here are 5 follow-up questions related to this topic:

1. What is the power rule for derivatives?
2. How do you differentiate a square root function?
3. Can constants affect the derivative of a function?
4. What happens to negative exponents when differentiating?
5. How do you handle products of variables when differentiating?

**Tip:** Always try to rewrite functions with radicals or fractional exponents to make differentiation easier!

Find the derivative of the following functions: f(x) = x^7, g(x) = 1/√x, h(x) = x^2 + 5x - 7, y = 4x^3 - 3x√x + (1/2)x, h(t) = 2/t√t - 3t.

Step-by-step solutions to finding derivatives using the power rule and simplifying radicals for functions such as f(x) = x^7, g(x) = 1/√x, h(x) = x^2 + 5x - 7, and more. Perfect for Calculus I students.

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